Your FREE AP Precalculus Practice Test 2026 – 230+ Q&A

AP Precalculus Practice Questions

1. A polynomial function f has a local maximum at $x = -2$ and a local minimum at $x = 5$, with no other extrema. What is the minimum possible degree of f?
   • Degree 2
   • Degree 3
   • Degree 5
   • Degree 4

   Correct answer: Degree 3

   Degree 3 is correct. Two turning points require at least degree 3, because a degree-n polynomial has at most $n - 1$ turning points, and $3 - 1 = 2$. A quadratic allows only one turning point, so degree 2 is too small.

2. Using the rational zeros test, which value is a possible rational zero of $f(x) = 2x^{3} + 3x - 4$?
   • $x = 3$
   • $x = 5$
   • $x = \frac{1}{3}$
   • $x = \frac{1}{2}$

   Correct answer: $x = \frac{1}{2}$

   $x = \frac{1}{2}$ is correct. The rational zeros theorem says any rational zero is a factor of the constant term 4 over a factor of the leading coefficient 2, giving candidates like $\pm 1, 2, 4, \frac{1}{2}$. The value $\frac{1}{2}$ fits this list, while $\frac{1}{3}$ and 5 do not because 3 and 5 are not factors of 2 or 4.

3. When $f(x) = x^{3} - 4x^{2} + x + 6$ is divided by $(x - 3)$ using synthetic division, what is the remainder?
   • $-3$
   • 9
   • 0
   • 6

   Correct answer: 0

   The remainder is 0. Evaluating $f(3) = 27 - 36 + 3 + 6 = 0$, and by the remainder theorem this equals the synthetic division remainder. A remainder of zero also confirms that $(x - 3)$ is a factor.

4. A rational function $r(x) = \frac{x^{2} - 4}{x^{2} + 1}$ has how many vertical asymptotes?
   • Zero
   • Two
   • One
   • Three

   Correct answer: Zero

   Zero is correct. Vertical asymptotes occur where the denominator equals zero for real inputs, but $x^{2} + 1 = 0$ has no real solutions since $x^{2}$ is never negative. Therefore the function has no vertical asymptotes even though the numerator has real zeros.

5. The function $f(x) = \frac{x^{2} + 2x - 3}{x - 1}$ can be simplified for $x \neq 1$. What is the simplified form?
   • $x - 1$
   • $x + 3$
   • $x - 3$
   • $x + 1$

   Correct answer: $x + 3$

   $x + 3$ is correct. The numerator factors as $(x + 3)(x - 1)$, and canceling the common $(x - 1)$ leaves $x + 3$, valid everywhere except $x = 1$ where a hole occurs. Choosing the wrong factor pair would give an incorrect simplification.

6. For the rational function $f(x) = \frac{3x^{2} - 5}{x - 2}$, polynomial long division gives a slant asymptote. What is that line?
   • $y = 3x + 6$
   • $y = 3x$
   • $y = 3x - 6$
   • $y = x + 2$

   Correct answer: $y = 3x + 6$

   $y = 3x + 6$ is correct. Dividing $3x^{2} - 5$ by $x - 2$ yields a quotient of $3x + 6$ with a nonzero remainder, and the slant asymptote is exactly that linear quotient. The remainder term shrinks to zero for large inputs, so the graph approaches this line.

7. What is the value of the polynomial $p(x) = x^{4} - 2x^{2}$ at $x = -1$?
   • 3
   • 1
   • $-3$
   • $-1$

   Correct answer: $-1$

   $p(-1) = -1$. Substituting gives $(-1)^{4} - 2(-1)^{2} = 1 - 2 \times 1 = 1 - 2 = -1$. Both even powers make the negative input positive before the subtraction.

8. A rational function $f(x) = \frac{x + 4}{x^{2} - 16}$ has a hole rather than an asymptote at which input?
   • $x = -4$
   • $x = 16$
   • $x = 4$
   • $x = 0$

   Correct answer: $x = -4$

   $x = -4$ is correct. The denominator factors as $(x - 4)(x + 4)$, so the $(x + 4)$ factor cancels with the numerator, leaving a removable discontinuity at $x = -4$. The factor $(x - 4)$ does not cancel, so $x = 4$ is a genuine vertical asymptote.

9. Descartes' rule of signs is applied to $f(x) = x^{3} - x^{2} + x - 1$. How many sign changes appear in $f(x)$?
   • Three
   • Two
   • Zero
   • One

   Correct answer: Three

   Three is correct. Reading the coefficient signs of $x^{3} - x^{2} + x - 1$ gives plus, minus, plus, minus, which switches sign three times. This count gives the maximum number of positive real zeros under Descartes' rule.

10. The graph of a polynomial passes through $(0, 0)$, $(1, 0)$, and $(-1, 0)$, each a simple crossing. Which is the lowest-degree polynomial consistent with this?
    • $f(x) = x(x - 1)^{2}$
    • $f(x) = x(x - 1)(x + 1)$
    • $f(x) = x^{2}(x - 1)$
    • $f(x) = (x - 1)(x + 1)$

    Correct answer: $f(x) = x(x - 1)(x + 1)$

    $f(x) = x(x - 1)(x + 1)$ is correct. Three simple crossings at 0, 1, and -1 require the three distinct factors $x$, $(x - 1)$, and $(x + 1)$, each to the first power for crossing behavior. Repeated factors would cause a bounce, and dropping a factor would miss a required zero.

11. For $f(x) = x^{2}$, what is the difference quotient $\frac{f(x + h) - f(x)}{h}$ simplified?
    • $2x + h$
    • $x + h$
    • $2xh$
    • $2x$

    Correct answer: $2x + h$

    $2x + h$ is correct. Expanding $f(x + h) = x^{2} + 2xh + h^{2}$ and subtracting $x^{2}$ leaves $2xh + h^{2}$, and dividing by h gives $2x + h$. This expression represents the average rate of change over an interval of width h.

12. A rational function has the equation $f(x) = \frac{2x^{2}}{x^{2} - 1}$. What is its horizontal asymptote?
    • $y = 0$
    • There is none
    • $y = 1$
    • $y = 2$

    Correct answer: $y = 2$

    $y = 2$ is correct. The numerator and denominator both have degree 2, so the horizontal asymptote is the ratio of leading coefficients, $\frac{2}{1} = 2$. Equal degrees rule out an asymptote of $y = 0$, which requires a smaller numerator degree.

13. If a polynomial f has $f(2) = 0$ and $f(-3) = 0$, which product must divide $f(x)$?
    • $(x + 2)(x - 3)$
    • $(x + 2)(x + 3)$
    • $(x - 2)(x + 3)$
    • $(x - 2)(x - 3)$

    Correct answer: $(x - 2)(x + 3)$

    $(x - 2)(x + 3)$ is correct. By the factor theorem, a zero at $x = 2$ gives the factor $(x - 2)$ and a zero at $x = -3$ gives the factor $(x + 3)$, so their product divides f. The other choices place the zeros at the wrong values.

14. A polynomial model for the volume of an open box is $V(x) = x(10 - 2x)(8 - 2x)$, where x is the cutout length. What is the domain that makes physical sense?
    • $0 < x < 4$
    • $x > 5$
    • All real numbers
    • $0 < x < 8$

    Correct answer: $0 < x < 4$

    $0 < x < 4$ is correct. The cutout x must be positive, and each side length $10 - 2x$ and $8 - 2x$ must stay positive, requiring $x < 5$ and $x < 4$ respectively. The tighter restriction $x < 4$ controls, so the meaningful domain is $0 < x < 4$.

15. What is the leading term of the product $(3x^{2})(x^{3} - 2x)$?
    • $-6x^{3}$
    • $x^{5}$
    • $3x^{5}$
    • $3x^{6}$

    Correct answer: $3x^{5}$

    $3x^{5}$ is correct. Multiplying the highest-degree terms $3x^{2}$ and $x^{3}$ gives $3 \times x^{2 + 3} = 3x^{5}$, which dominates end behavior. The term $-6x^{3}$ is a lower-degree product, not the leading term.

16. The graph of a rational function flattens toward $y = 0$ on both ends but spikes to infinity near $x = 4$. Which description fits?
    • Slant asymptote and a hole at $x = 4$
    • Two horizontal asymptotes
    • No asymptotes at all
    • Horizontal asymptote $y = 0$ and a vertical asymptote at $x = 4$

    Correct answer: Horizontal asymptote $y = 0$ and a vertical asymptote at $x = 4$

    Horizontal asymptote $y = 0$ with a vertical asymptote at $x = 4$ is correct. Flattening toward zero on both ends indicates the horizontal asymptote $y = 0$, and the unbounded spike near $x = 4$ marks a vertical asymptote there. A hole would not cause the output to spike to infinity.

17. A degree 4 polynomial with positive leading coefficient has exactly two real zeros, both simple. How many times does its graph touch but not cross the x-axis?
    • Two times
    • One time
    • Zero times
    • Four times

    Correct answer: Zero times

    Zero times is correct. Both real zeros are simple, meaning odd multiplicity, so the graph crosses at each rather than touching and turning. Touching without crossing requires even multiplicity, which is not present here.

18. When two polynomials of degrees 3 and 2 are added, what is the degree of the resulting polynomial?
    • 2
    • 3
    • 5
    • 6

    Correct answer: 3

    Degree 3 is correct. Adding polynomials keeps the highest degree present, and since 3 is larger than 2, the sum has degree 3 as long as the degree-3 term is not canceled. Degrees are multiplied, not added, only when multiplying polynomials.

19. A rational function $f(x) = \frac{x^{3}}{x^{2} + 1}$ has what type of end behavior asymptote?
    • A horizontal asymptote $y = 0$
    • A horizontal asymptote $y = 1$
    • A slant asymptote $y = x$
    • No asymptote

    Correct answer: A slant asymptote $y = x$

    A slant asymptote $y = x$ is correct. The numerator degree 3 is exactly one more than the denominator degree 2, so division gives a linear quotient of $y = x$ that the graph approaches. A horizontal asymptote would require the degrees to be equal or the numerator degree to be smaller.

20. The polynomial $f(x) = 6x^{2} - 13x + 6$ factors completely as which product?
    • $(2x - 3)(3x - 2)$
    • $(2x - 3)(3x + 2)$
    • $(6x - 1)(x - 6)$
    • $(2x + 3)(3x + 2)$

    Correct answer: $(2x - 3)(3x - 2)$

    $(2x - 3)(3x - 2)$ is correct. Expanding gives $6x^{2} - 4x - 9x + 6 = 6x^{2} - 13x + 6$, matching the original. Both constants are negative to produce the positive constant 6 and negative middle term.

21. A polynomial $p(x)$ is divided by $(x + 2)$ and leaves a nonzero remainder of $-5$. What is $p(-2)$?
    • 5
    • 0
    • 2
    • $-5$

    Correct answer: $-5$

    $p(-2) = -5$. The remainder theorem states the remainder upon division by $(x + 2)$, which is $(x - (-2))$, equals $p(-2)$. Since that remainder is $-5$, $p(-2) = -5$, and a nonzero remainder confirms $(x + 2)$ is not a factor.

22. A polynomial function is even and has a zero at $x = 3$. Which other value must also be a zero?
    • $x = 0$
    • $x = -3$
    • $x = \frac{1}{3}$
    • $x = 6$

    Correct answer: $x = -3$

    $x = -3$ is correct. An even function satisfies $f(-x) = f(x)$, so $f(-3) = f(3) = 0$, forcing a matching zero at $x = -3$. The symmetry of even functions pairs zeros at opposite inputs.

23. For the function $f(x) = \frac{5}{x^{2} + 4x + 4}$, what is the location and nature of the discontinuity?
    • A hole at $x = -2$
    • A vertical asymptote at $x = -2$
    • A vertical asymptote at $x = 2$
    • No discontinuity

    Correct answer: A vertical asymptote at $x = -2$

    A vertical asymptote at $x = -2$ is correct. The denominator factors as $(x + 2)^{2}$, which is zero at $x = -2$, and since the numerator 5 is nonzero there, the function blows up to infinity, giving a vertical asymptote. There is no canceling factor, so it is not a hole.

24. A polynomial of degree 7 with real coefficients has 3 distinct non-real zeros listed. Why is this impossible?
    • All zeros of a degree 7 polynomial are real
    • A polynomial cannot have distinct zeros
    • Degree 7 polynomials have no complex zeros
    • Non-real zeros must come in conjugate pairs, so the count must be even

    Correct answer: Non-real zeros must come in conjugate pairs, so the count must be even

    Conjugate pairs require an even count, which is correct. For real-coefficient polynomials, each non-real zero is paired with its conjugate, so non-real zeros always total an even number. Three is odd, making the listed situation impossible.

25. A polynomial passes through the point $(0, -8)$ and has no other information given. What is its constant term?
    • $-8$
    • Cannot be determined
    • 0
    • 8

    Correct answer: $-8$

    The constant term is $-8$. The y-intercept of a polynomial equals its constant term, found by evaluating at $x = 0$, and the point $(0, -8)$ gives $f(0) = -8$ directly. So the constant term must be $-8$.

26. The graph of $f(x) = x^{3} - x$ is rewritten in factored form. Which factorization is correct?
    • $x(x^{2} + 1)$
    • $x^{2}(x - 1)$
    • $(x - 1)(x + 1)$
    • $x(x - 1)(x + 1)$

    Correct answer: $x(x - 1)(x + 1)$

    $x(x - 1)(x + 1)$ is correct. Factoring out x gives $x(x^{2} - 1)$, and the difference of squares $x^{2} - 1$ factors into $(x - 1)(x + 1)$. This reveals three simple real zeros at 0, 1, and -1.

27. A rational function model for drug concentration is $C(t) = \frac{50t}{t^{2} + 9}$. As t increases without bound, what does C approach?
    • 9
    • 0
    • Infinity
    • 50

    Correct answer: 0

    C approaches 0. The denominator degree 2 exceeds the numerator degree 1, so the horizontal asymptote is $y = 0$ and concentration tends toward zero for large times. This models a drug clearing from the system over time.

28. What is the standard form expansion of $(x - 4)(x + 1)$?
    • $x^{2} + 3x - 4$
    • $x^{2} - 5x - 4$
    • $x^{2} - 3x - 4$
    • $x^{2} - 4x + 1$

    Correct answer: $x^{2} - 3x - 4$

    $x^{2} - 3x - 4$ is correct. Multiplying gives $x^{2} + x - 4x - 4$, and combining the middle terms yields $x^{2} - 3x - 4$. The constant $-4$ comes from multiplying $-4$ by 1.

29. A function f is concave down on $(-\infty, 1)$ and concave up on $(1, \infty)$. What occurs at $x = 1$?
    • A point of inflection
    • A removable discontinuity
    • A vertical asymptote
    • A relative maximum

    Correct answer: A point of inflection

    A point of inflection is correct. Concavity changing from down to up at $x = 1$ defines an inflection point, where the bending direction of the graph reverses. This is unrelated to extrema or to asymptotes, which involve direction changes or undefined behavior instead.

30. A rational function $r(x) = \frac{x - 3}{x^{2} - 9}$ is simplified. What is the simplified form and excluded value besides $x = -3$?
    • $\frac{1}{x - 3}$, with $x \neq 3$
    • $x - 3$, with no restriction
    • $\frac{1}{x + 3}$, with $x \neq 3$
    • $(x + 3)$, with $x \neq 3$

    Correct answer: $\frac{1}{x + 3}$, with $x \neq 3$

    $\frac{1}{x + 3}$ with $x \neq 3$ is correct. The denominator factors as $(x - 3)(x + 3)$, and canceling the common $(x - 3)$ leaves $\frac{1}{x + 3}$, creating a hole at $x = 3$. The remaining factor $(x + 3)$ still gives a vertical asymptote at $x = -3$.

31. A degree 3 polynomial has end behavior falling on the left and rising on the right. What is true about its leading coefficient?
    • It cannot be determined from end behavior
    • It is zero
    • It is negative
    • It is positive

    Correct answer: It is positive

    Positive is correct. An odd-degree polynomial that falls on the left and rises on the right matches a positive leading coefficient, the same pattern as the basic cubic. A negative leading coefficient would reverse the ends to rise on the left and fall on the right.

32. Which polynomial has exactly the zeros $x = 0$ (multiplicity 1), $x = 2$ (multiplicity 2), and degree 3?
    • $f(x) = x(x - 2)$
    • $f(x) = x(x - 2)^{2}$
    • $f(x) = x(x + 2)^{2}$
    • $f(x) = x^{2}(x - 2)$

    Correct answer: $f(x) = x(x - 2)^{2}$

    $f(x) = x(x - 2)^{2}$ is correct. The factor x gives a simple zero at 0, and $(x - 2)^{2}$ gives a double zero at 2, and the total degree $1 + 2 = 3$ matches. The other forms misplace the multiplicities or the zero locations.

33. The average rate of change of $f(x) = x^{3}$ over the interval from $x = 0$ to $x = 2$ is what value?
    • 8
    • 2
    • 4
    • 6

    Correct answer: 4

    The answer is 4. The average rate of change is $\frac{f(2) - f(0)}{2 - 0} = \frac{8 - 0}{2} = 4$. Forgetting to divide by the interval width of 2 would incorrectly give 8.

34. A rational function has numerator $(x^{2} + 1)$, which has no real zeros, and denominator $(x - 2)$. What does the lack of real numerator zeros tell you about the graph?
    • The graph has no x-intercepts
    • The graph has a hole at $x = 2$
    • The graph has two x-intercepts
    • The graph has no vertical asymptote

    Correct answer: The graph has no x-intercepts

    No x-intercepts is correct. An x-intercept requires the numerator to be zero for a real input, but $x^{2} + 1$ is never zero for real x, so the graph never crosses the x-axis. The denominator zero at $x = 2$ still produces a vertical asymptote.

35. When dividing $f(x) = 2x^{2} + 3x - 2$ by $(x + 2)$, what is the quotient?
    • $2x + 1$
    • $x - 1$
    • $2x - 1$
    • $2x - 2$

    Correct answer: $2x - 1$

    $2x - 1$ is correct. Since $f(-2) = 8 - 6 - 2 = 0$, $(x + 2)$ is a factor, and dividing gives the quotient $2x - 1$ because $(x + 2)(2x - 1) = 2x^{2} + 3x - 2$. The product check confirms the quotient.

36. A polynomial model of profit $P(x) = -2(x - 30)(x - 70)$ describes profit by number of units x. At which x values is profit zero?
    • $x = 0$ and $x = 100$
    • $x = -30$ and $x = -70$
    • $x = 2$ and $x = 50$
    • $x = 30$ and $x = 70$

    Correct answer: $x = 30$ and $x = 70$

    $x = 30$ and $x = 70$ is correct. Setting each factor equal to zero gives $x = 30$ and $x = 70$, the break-even points where profit equals zero. The leading factor $-2$ affects the height of the parabola but not the locations of these zeros.

37. The reciprocal-squared function $f(x) = \frac{1}{x^{2}}$ has what end behavior and sign?
    • It approaches $y = 1$ on both ends
    • It approaches $y = 0$ from above on both ends
    • It grows without bound on both ends
    • It approaches $y = 0$ from below on both ends

    Correct answer: It approaches $y = 0$ from above on both ends

    Approaching $y = 0$ from above is correct. Squaring keeps the denominator positive for all nonzero inputs, so outputs stay positive, and for large magnitude inputs $\frac{1}{x^{2}}$ shrinks toward zero. Thus the graph nears the horizontal asymptote $y = 0$ from above on both sides.

38. A polynomial f has a graph that crosses the x-axis at $x = -2$, bounces at $x = 1$, and crosses at $x = 4$. What is the smallest possible degree?
    • 5
    • 4
    • 2
    • 3

    Correct answer: 4

    Degree 4 is correct. Crossing at -2 and 4 each requires odd multiplicity (at least 1 each), and bouncing at 1 requires even multiplicity (at least 2), giving a minimum total of $1 + 2 + 1 = 4$. A lower degree could not provide the bounce plus two crossings.

39. A polynomial $g(x) = (x^{2} + 5)$ has what kind of real zeros?
    • One real zero
    • Three real zeros
    • Two real zeros
    • No real zeros

    Correct answer: No real zeros

    No real zeros is correct. Setting $x^{2} + 5 = 0$ gives $x^{2} = -5$, which has no real solution since a square is never negative. The zeros of this expression are non-real complex numbers, so the graph never touches the x-axis.

40. A polynomial f has zeros at $x = -2$, $x = 0$, and $x = 3$, each of multiplicity 1, and a positive leading coefficient with degree 3. What is the sign of $f(x)$ on the interval between $x = 0$ and $x = 3$?
    • Positive throughout the interval
    • Negative throughout the interval
    • Zero throughout the interval
    • Positive then negative within the interval

    Correct answer: Negative throughout the interval

    Negative throughout is correct. With simple zeros at -2, 0, and 3 and a positive leading coefficient, the cubic is positive for $x > 3$, then alternates sign at each crossing, making it negative on $(0, 3)$. A single-multiplicity zero produces one sign change, so the function does not switch sign again inside the interval.

41. A polynomial $g(x)$ is the product $(x - 1)(x + 4)(x - 1)$. Which statement correctly describes its behavior at $x = 1$?
    • The graph crosses the x-axis steeply at $x = 1$
    • The graph touches the x-axis at $x = 1$ and turns back
    • The graph has a vertical asymptote at $x = 1$
    • The graph has a hole at $x = 1$

    Correct answer: The graph touches the x-axis at $x = 1$ and turns back

    Touching and turning is correct. The factor $(x - 1)$ appears twice, giving $x = 1$ a multiplicity of 2, which is even, so the graph touches the axis there and reverses without crossing. Holes and asymptotes are rational-function features, not polynomial-zero behavior.

42. For the rational function $f(x) = \frac{x + 3}{x^{2} + x - 6}$, at which input is there a removable discontinuity?
    • $x = 2$
    • $x = -3$
    • $x = -2$
    • $x = 6$

    Correct answer: $x = -3$

    $x = -3$ is correct. The denominator factors as $(x + 3)(x - 2)$, so the numerator $(x + 3)$ cancels with one denominator factor, removing the discontinuity at $x = -3$ and leaving a hole there. The remaining $(x - 2)$ factor does not cancel, so $x = 2$ is a vertical asymptote, not a hole.

43. Which polynomial has end behavior in which both ends rise and exactly two distinct real zeros?
    • $f(x) = -(x - 1)(x + 1)$
    • $f(x) = (x - 1)^{2} (x + 1)^{2}$
    • $f(x) = (x - 1)(x + 1)(x - 2)$
    • $f(x) = -(x - 1)^{2} (x + 1)^{2}$

    Correct answer: $f(x) = (x - 1)^{2} (x + 1)^{2}$

    The fourth-degree product $(x - 1)^{2} (x + 1)^{2}$ is correct. Its degree is 4 (even) with a positive leading coefficient, so both ends rise, and it has exactly two distinct real zeros at 1 and -1 (each multiplicity 2). The negative versions send both ends down, and the cubic has opposite end behaviors.

44. A rational function $h(x) = \frac{x^{2} - 4}{x + 2}$ is simplified. What is the resulting expression and its restriction?
    • $x - 2$, with $x \neq -2$
    • $x + 2$, with $x \neq 2$
    • $x - 2$, with no restriction
    • $x + 2$, with $x \neq -2$

    Correct answer: $x - 2$, with $x \neq -2$

    $x - 2$ with $x \neq -2$ is correct. The numerator is a difference of squares, $(x - 2)(x + 2)$; canceling the common $(x + 2)$ leaves $x - 2$. The original function is undefined at $x = -2$, so that input is excluded, leaving a hole there.

45. A polynomial of degree 4 has a positive leading coefficient. What is the maximum number of times its graph can intersect a horizontal line?
    • 2
    • 4
    • 3
    • 5

    Correct answer: 4

    Four is correct. Setting the polynomial equal to a constant gives another degree-4 equation, which can have at most 4 real solutions, so the graph can meet any horizontal line at most 4 times. The count cannot exceed the degree.

46. Given $f(x) = 3x^{2}$ and the table values $f(1) = 3$, $f(2) = 12$, $f(3) = 27$, what are the first differences of the outputs over these equally spaced inputs?
    • 3 and 12
    • 9 and 15
    • 6 and 6
    • 12 and 27

    Correct answer: 9 and 15

    9 and 15 is correct. The first differences subtract consecutive outputs: $12 - 3 = 9$ and $27 - 12 = 15$. These are not constant, which is expected for a quadratic, whose second differences $(15 - 9 = 6)$ are the constant ones instead.

47. A rational function $r(x)$ has a numerator of degree 2 and a denominator of degree 5. Which describes its end behavior?
    • It approaches a slant asymptote
    • It increases without bound on both ends
    • It approaches the horizontal asymptote $y = 0$
    • It has no asymptotes

    Correct answer: It approaches the horizontal asymptote $y = 0$

    Approaching $y = 0$ is correct. Because the denominator degree (5) exceeds the numerator degree (2), the denominator dominates for large inputs and the outputs shrink toward zero. A slant asymptote would require the numerator degree to exceed the denominator degree by exactly one.

48. A polynomial $p(x)$ yields $p(-1) = 0$. Which factor must appear in its factored form?
    • $(x - 1)$
    • $(x + 1)$
    • $(x + 0)$
    • $(1 - x)$

    Correct answer: $(x + 1)$

    $(x + 1)$ is correct. By the Factor Theorem, a zero at $x = c$ corresponds to the factor $(x - c)$; here $c = -1$, so the factor is $(x - (-1)) = (x + 1)$. The factor $(x - 1)$ would correspond to a zero at $x = 1$ instead.

49. The graph of a function rises to a peak, falls to a valley, rises to another peak, and then falls. What is the minimum polynomial degree consistent with this shape?
    • Degree 3
    • Degree 4
    • Degree 2
    • Degree 5

    Correct answer: Degree 4

    Degree 4 is correct. The described shape has three turning points (peak, valley, peak), and a degree-n polynomial allows at most $n - 1$ turning points, so three turning points require at least degree 4. A cubic permits only two turning points.

50. Which expression is a fully factored polynomial with a zero of multiplicity 3 at $x = 4$ and no other real zeros?
    • $(x - 4)^{3}$
    • $(x + 4)^{3}$
    • $(x - 4)^{2}$
    • $(x - 4)(x^{2} + 1)$

    Correct answer: $(x - 4)^{3}$

    $(x - 4)^{3}$ is correct. The factor $(x - 4)$ raised to the third power gives a single real zero at $x = 4$ with multiplicity 3 and no other real zeros. The version with $(x + 4)$ places the zero at -4, and the squared form has multiplicity 2.

51. For $f(x) = \frac{x - 3}{x^{2} - 9}$, what feature occurs at $x = 3$?
    • A vertical asymptote
    • An x-intercept
    • A horizontal asymptote
    • A hole

    Correct answer: A hole

    A hole is correct. The denominator factors as $(x - 3)(x + 3)$, and the $(x - 3)$ in the numerator cancels with one in the denominator, removing the discontinuity at $x = 3$ and leaving a hole. The remaining $(x + 3)$ factor produces a vertical asymptote at $x = -3$ instead.

52. A polynomial model $f(t)$ gives the height of a drone in meters at time $t$ seconds. If $f$ has a relative minimum at $(6, 10)$, what does this mean in context?
    • The drone is highest at 6 seconds
    • The drone reaches a local low of 10 meters at 6 seconds before rising again
    • The drone is on the ground at 6 seconds
    • The drone descends forever after 6 seconds

    Correct answer: The drone reaches a local low of 10 meters at 6 seconds before rising again

    A local low of 10 meters at 6 seconds is correct. A relative minimum is a point where the output is smaller than at nearby inputs, so the drone dips to 10 meters at 6 seconds and then climbs. It is not the overall lowest point nor ground level unless the height there were zero.

53. Which statement about the graph of an odd-degree polynomial with a negative leading coefficient is correct?
    • The left end falls and the right end rises
    • Both ends fall
    • Both ends rise
    • The left end rises and the right end falls

    Correct answer: The left end rises and the right end falls

    Left rises and right falls is correct. Odd degree gives opposite end behaviors, and a negative leading coefficient makes the right end fall while the left end rises. Both ends going the same way would require an even degree.

54. A rational function $f(x) = \frac{4x^{2} - 1}{2x^{2} + 3}$ has what horizontal asymptote?
    • $y = 4$
    • $y = 0$
    • $y = 2$
    • There is no horizontal asymptote

    Correct answer: $y = 2$

    $y = 2$ is correct. The numerator and denominator both have degree 2, so the horizontal asymptote is the ratio of leading coefficients, $\frac{4}{2} = 2$. A value of $y = 0$ would require the numerator degree to be smaller than the denominator degree.

55. If a polynomial has the factored form $f(x) = (x - 5)(x + 2)^{3}$, what is the total degree of $f$?
    • 5
    • 3
    • 4
    • 2

    Correct answer: 4

    Degree 4 is correct. The total degree is the sum of the exponents on the factors: $(x - 5)$ contributes 1 and $(x + 2)^{3}$ contributes 3, giving $1 + 3 = 4$. The degree counts multiplicities, not just the number of distinct factors.

56. A polynomial passes through $(0, -6)$ and has the form $f(x) = a(x - 1)(x + 3)$. What is the value of $a$?
    • 2
    • -2
    • 6
    • -6

    Correct answer: 2

    $a = 2$ is correct. Substituting $x = 0$ gives $f(0) = a(0 - 1)(0 + 3) = a \times (-1) \times 3 = -3a$, and setting this equal to -6 yields $-3a = -6$, so $a = 2$. The other values do not satisfy the point.

57. The graph of a rational function has vertical asymptotes at $x = -1$ and $x = 4$. Which denominator is consistent with this?
    • $(x - 1)(x + 4)$
    • $(x + 1)(x - 4)$
    • $(x + 1)(x + 4)$
    • $(x - 1)(x - 4)$

    Correct answer: $(x + 1)(x - 4)$

    $(x + 1)(x - 4)$ is correct. Vertical asymptotes occur where the denominator is zero and does not cancel; setting $(x + 1) = 0$ gives $x = -1$ and $(x - 4) = 0$ gives $x = 4$, matching the asymptotes. The other denominators place the zeros at the wrong values.

58. A polynomial $f$ satisfies $f(2) = 5$ and $f(6) = 13$. What is its average rate of change on the interval from $x = 2$ to $x = 6$?
    • 8
    • 4
    • 2
    • 3

    Correct answer: 2

    The average rate of change is 2. It equals $\frac{f(6) - f(2)}{6 - 2} = \frac{13 - 5}{4} = \frac{8}{4} = 2$. Forgetting to divide by the interval width of 4 would give the incorrect value of 8.

59. Which describes the relationship between the factors of a polynomial and the x-intercepts where the graph crosses versus touches the axis?
    • Even-multiplicity factors cause crossing; odd-multiplicity factors cause touching
    • Odd-multiplicity factors cause crossing; even-multiplicity factors cause touching
    • All factors cause the graph to cross
    • All factors cause the graph to touch and turn

    Correct answer: Odd-multiplicity factors cause crossing; even-multiplicity factors cause touching

    Odd causes crossing and even causes touching is correct. A factor with odd multiplicity makes the graph pass through the axis, while a factor with even multiplicity makes it touch and turn back. The behavior depends on whether the multiplicity is odd or even.

60. A rational function $f(x) = \frac{x^{2} + 1}{x - 2}$ has numerator degree one more than denominator degree. What type of asymptote does it have besides the vertical one?
    • A horizontal asymptote at $y = 1$
    • A slant asymptote
    • A second vertical asymptote
    • No additional asymptote

    Correct answer: A slant asymptote

    A slant asymptote is correct. Since the numerator degree (2) exceeds the denominator degree (1) by exactly one, dividing gives a linear quotient that the graph approaches, producing a slant asymptote. There is no horizontal asymptote when the numerator degree is larger.

61. If a polynomial of degree 7 has 5 distinct real zeros and one of them has multiplicity 3 while the rest are simple, how many non-real zeros does it have, counted with multiplicity?
    • 0
    • 2
    • 4
    • 1

    Correct answer: 0

    Zero non-real zeros is correct. The real zeros account for 3 (from the triple) plus 4 simple zeros $= 7$ total with multiplicity, which equals the degree. Since all 7 zeros are real, none are non-real.

62. The function $f(x) = x^{3}$ is transformed into $g(x) = \frac{1}{2} \times x^{3}$. How does this affect the graph?
    • A vertical compression by a factor of $\frac{1}{2}$
    • A horizontal stretch by a factor of $\frac{1}{2}$
    • A vertical stretch by a factor of 2
    • A reflection over the x-axis

    Correct answer: A vertical compression by a factor of $\frac{1}{2}$

    A vertical compression by $\frac{1}{2}$ is correct. Multiplying the output by a constant between 0 and 1 shrinks every y-value toward the x-axis, compressing the graph vertically. A factor greater than 1 would stretch it instead, and a negative factor would add a reflection.

63. A polynomial model for a company's monthly profit has a y-intercept of -8 (in thousands of dollars). What does this represent in context?
    • The break-even production level
    • The maximum monthly profit
    • The profit when no units are produced, an 8 thousand dollar loss
    • The number of units that maximize profit

    Correct answer: The profit when no units are produced, an 8 thousand dollar loss

    An 8 thousand dollar loss at zero production is correct. The y-intercept is the output when the input is zero, so it gives the profit when no units are made, here a loss of 8 thousand dollars from fixed costs. Break-even points are zeros, not the y-intercept.

64. Which polynomial is written in standard form?
    • $f(x) = 3 + 2x^{2} - x$
    • $f(x) = -x + 3 + 2x^{2}$
    • $f(x) = 2x^{2} - x + 3$
    • $f(x) = 2x^{2} + 3 - x$

    Correct answer: $f(x) = 2x^{2} - x + 3$

    $2x^{2} - x + 3$ is correct. Standard form lists terms in order of decreasing exponent, here the squared term, then the linear term, then the constant. The other expressions contain the same terms but are not arranged from highest to lowest degree.

65. A rational function $f(x) = \frac{6}{x^{2} + 2}$ has what kind of vertical asymptotes?
    • One vertical asymptote at $x = 0$
    • No vertical asymptotes
    • Two vertical asymptotes
    • A vertical asymptote at $x = 2$

    Correct answer: No vertical asymptotes

    No vertical asymptotes is correct. The denominator $x^{2} + 2$ is always at least 2 and never equals zero for any real input, so the function is defined everywhere and has no vertical asymptotes. Vertical asymptotes require real zeros of the denominator.

66. Two equally spaced data sets are compared. Set A has constant first differences and Set B has constant second differences. Which polynomial degrees do they suggest?
    • Set A linear, Set B quadratic
    • Set A quadratic, Set B linear
    • Set A cubic, Set B quadratic
    • Both linear

    Correct answer: Set A linear, Set B quadratic

    Set A linear and Set B quadratic is correct. Constant first differences indicate a degree-1 (linear) pattern, and constant second differences indicate a degree-2 (quadratic) pattern. The order of differences that becomes constant matches the polynomial degree.

67. The polynomial $f(x) = -(x + 1)(x - 2)(x - 5)$ is evaluated for very large negative $x$. What is the end behavior on the left?
    • $f(x)$ rises toward positive infinity
    • $f(x)$ approaches a horizontal asymptote
    • $f(x)$ approaches zero
    • $f(x)$ falls toward negative infinity

    Correct answer: $f(x)$ rises toward positive infinity

    Rising on the left is correct. Expanding gives a leading term of $-x^{3}$ (odd degree, negative leading coefficient), so the left end rises while the right end falls. For very large negative $x$, the negative coefficient times a large negative cube yields large positive outputs.

68. A polynomial graph touches the x-axis at $x = 0$, crosses at $x = 3$, and touches at $x = -4$. What is the minimum possible degree?
    • 3
    • 5
    • 4
    • 6

    Correct answer: 5

    Degree 5 is correct. Touching at $x = 0$ requires multiplicity at least 2, crossing at $x = 3$ requires multiplicity at least 1, and touching at $x = -4$ requires multiplicity at least 2, summing to $2 + 1 + 2 = 5$. This is the smallest total degree consistent with the described behavior.

69. Which transformation of $f(x) = x^{2}$ produces a parabola with vertex at $(3, -1)$ that opens upward?
    • $g(x) = (x - 3)^{2} - 1$
    • $g(x) = (x + 3)^{2} - 1$
    • $g(x) = (x - 3)^{2} + 1$
    • $g(x) = -(x - 3)^{2} - 1$

    Correct answer: $g(x) = (x - 3)^{2} - 1$

    $g(x) = (x - 3)^{2} - 1$ is correct. The $(x - 3)$ shifts the vertex right to $x = 3$, and the minus 1 shifts it down to $y = -1$, while the positive coefficient keeps it opening upward. The negative version would open downward, and $(x + 3)$ would move the vertex left.

70. A rational function $f(x) = \frac{3x}{x - 2}$ has what horizontal asymptote?
    • $y = 3$
    • $y = 0$
    • $y = 2$
    • There is no horizontal asymptote

    Correct answer: $y = 3$

    $y = 3$ is correct. The numerator and denominator both have degree 1, so the horizontal asymptote is the ratio of leading coefficients, $\frac{3}{1} = 3$. The function levels off toward $y = 3$ for large magnitude inputs.

71. If a polynomial has real coefficients and exactly the zeros 2, $2i$, and $-2i$, what is its minimum possible degree?
    • 2
    • 1
    • 4
    • 3

    Correct answer: 3

    Degree 3 is correct. The real zero 2 contributes one factor, and the non-real zeros $2i$ and $-2i$ form a conjugate pair contributing two more, giving a minimum of $1 + 2 = 3$. Conjugate pairs are required because the coefficients are real.

72. A polynomial $f$ is decreasing on the interval $(1, 5)$. What can be said about its average rate of change over that interval?
    • It is positive
    • It is zero
    • It is negative
    • It cannot be determined

    Correct answer: It is negative

    Negative is correct. If a function is decreasing across an entire interval, its output at the right endpoint is lower than at the left endpoint, so the net change is negative and the average rate of change is negative. A decreasing function cannot have a positive average rate of change over the interval.

73. Which polynomial has a graph that is symmetric about the origin?
    • $f(x) = x^{3} - 4x$
    • $f(x) = x^{2} + 1$
    • $f(x) = x^{3} + 2$
    • $f(x) = x^{4}$

    Correct answer: $f(x) = x^{3} - 4x$

    $f(x) = x^{3} - 4x$ is correct. Both terms have odd exponents and there is no constant term, so the function is odd and its graph is symmetric about the origin. The other options contain even-power terms or a nonzero constant, which break origin symmetry.

74. A rational function $f(x) = \frac{(x - 1)(x + 2)}{(x + 2)(x - 4)}$ is graphed. Which features appear at $x = -2$ and $x = 4$ respectively?
    • Vertical asymptote at -2 and hole at 4
    • Hole at -2 and vertical asymptote at 4
    • Holes at both -2 and 4
    • Vertical asymptotes at both -2 and 4

    Correct answer: Hole at -2 and vertical asymptote at 4

    Hole at -2 and vertical asymptote at 4 is correct. The factor $(x + 2)$ cancels between numerator and denominator, producing a hole at $x = -2$, while the non-canceling $(x - 4)$ leaves the denominator zero at $x = 4$, producing a vertical asymptote there.

75. A polynomial model for the volume of a box is $V(x) = x(10 - 2x)(8 - 2x)$, where $x$ is the height. For what positive inputs is the volume model physically meaningful (positive volume)?
    • $0 < x < 4$
    • $0 < x < 5$
    • $x > 5$
    • All positive $x$

    Correct answer: $0 < x < 4$

    $0 < x < 4$ is correct. Volume must be positive, which requires all three factors positive: $x > 0$, $10 - 2x > 0$ (so $x < 5$), and $8 - 2x > 0$ (so $x < 4$). The most restrictive bound gives $0 < x < 4$, where every dimension is positive.

76. What is the result of dividing the polynomial $x^{2} - x - 6$ by $(x - 3)$?
    • $x + 2$
    • $x - 2$
    • $x + 3$
    • $x - 6$

    Correct answer: $x + 2$

    $x + 2$ is correct. The dividend factors as $(x - 3)(x + 2)$, so dividing by $(x - 3)$ leaves the quotient $x + 2$ with no remainder. This also confirms that $x = 3$ is a zero of the original polynomial.

77. A polynomial of even degree has a positive leading coefficient and a negative value at its lowest turning point. How many real zeros must it have at minimum?
    • Exactly 1
    • At least 2
    • Exactly 0
    • Exactly the degree

    Correct answer: At least 2

    At least 2 is correct. With both ends rising (positive even-degree) and a turning point dipping below the x-axis, the graph must cross the axis on the way down and again on the way back up, giving at least two real zeros. An even-degree polynomial that dips below zero cannot have zero real zeros.

78. The reciprocal-squared function $f(x) = \frac{1}{x^{2}}$ differs from $f(x) = \frac{1}{x}$ in which way near its vertical asymptote at $x = 0$?
    • $\frac{1}{x^{2}}$ goes to positive infinity on both sides, while $\frac{1}{x}$ goes to opposite infinities
    • Both go to negative infinity on both sides
    • $\frac{1}{x^{2}}$ goes to opposite infinities on each side
    • Both behave identically near zero

    Correct answer: $\frac{1}{x^{2}}$ goes to positive infinity on both sides, while $\frac{1}{x}$ goes to opposite infinities

    $\frac{1}{x^{2}}$ rising on both sides is correct. Squaring the denominator keeps outputs positive near zero, so $\frac{1}{x^{2}}$ tends to positive infinity from both sides, whereas $\frac{1}{x}$ is positive on one side and negative on the other, giving opposite infinities. The even power changes the sign behavior near the asymptote.

79. A polynomial $p(x)$ has $p(0) = 4$ and is known to be even. Which point must also lie on its graph?
    • $(-4, 0)$
    • $(4, 0)$
    • There is not enough information
    • $(0, -4)$

    Correct answer: There is not enough information

    Not enough information is correct. Knowing $p(0) = 4$ fixes only the single point $(0, 4)$, and evenness gives symmetry about the y-axis but does not by itself determine the output at any other specific input. No additional point can be guaranteed from the y-intercept and symmetry alone.

80. Which expression is the complete factorization of $x^{3} - 8$?
    • $(x - 2)(x^{2} + 2x + 4)$
    • $(x - 2)(x^{2} - 2x + 4)$
    • $(x - 2)(x^{2} - 4)$
    • $(x + 2)(x^{2} - 2x + 4)$

    Correct answer: $(x - 2)(x^{2} + 2x + 4)$

    The factorization $(x - 2)(x^{2} + 2x + 4)$ is correct. As a difference of cubes, $x^{3} - 2^{3}$ factors into $(x - 2)$ times the quadratic $x^{2} + 2x + 4$, using the pattern $a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})$. The choice with a plus sign in the linear factor or the wrong sign in the quadratic does not match the difference-of-cubes pattern.

81. When the polynomial $p(x) = x^{3} - 2x^{2} + x - 5$ is divided by $(x - 2)$, what is the remainder?
    • 5
    • 0
    • -3
    • -5

    Correct answer: -3

    The remainder is -3. By the Remainder Theorem the remainder equals $p(2)$, and substituting gives 8 minus 8 plus 2 minus 5 = -3. Computing $p(2)$ directly avoids carrying out the full division, and the nonzero result confirms that $(x - 2)$ is not a factor.

82. A polynomial with real coefficients has degree 4 and zeros at $x = 1$, $x = 1$, and $x = 2i$. What is its remaining zero?
    • $x = 1$
    • $x = -2i$
    • $x = 2$
    • $x = -1$

    Correct answer: $x = -2i$

    The remaining zero is $x = -2i$. A degree-4 polynomial has four zeros counted with multiplicity, and three are accounted for by the double zero at $x = 1$ and the single zero at $x = 2i$. Because real coefficients force non-real zeros into conjugate pairs, the partner of $2i$ must be its conjugate $-2i$.

83. Algebraically, which test confirms that $f(x) = x^{3} - 4x$ is an odd function?
    • Showing $f(-x) = f(x)$ for all x
    • Showing $f(-x) = -f(x)$, since $f(-x) = -x^{3} + 4x = -(x^{3} - 4x)$
    • Showing every exponent is even
    • Showing $f(0) = 0$

    Correct answer: Showing $f(-x) = -f(x)$, since $f(-x) = -x^{3} + 4x = -(x^{3} - 4x)$

    Showing $f(-x) = -f(x)$ is correct. Substituting $-x$ gives $(-x)^{3}$ minus $4(-x) = -x^{3}$ plus 4x, which is exactly the opposite of the original $f(x) = x^{3} - 4x$, confirming the function is odd. The condition $f(-x) = f(x)$ would instead define an even function, and $f(0) = 0$ alone does not establish odd symmetry.

84. For $f(x) = x^{2} + x$, the average rate of change on $[0, 2]$ compared to the average rate of change on $[2, 4]$ is which of the following?
    • Smaller, because the function is concave up
    • Larger, because the function is concave up
    • Equal, because the function is a polynomial
    • Smaller, because the function is concave down

    Correct answer: Smaller, because the function is concave up

    Smaller is correct. On $[0, 2]$ the rate equals [6 minus 0] over 2 = 3, while on $[2, 4]$ it equals [20 minus 6] over 2 = 7, so the earlier interval has the smaller rate. This rising sequence of average rates of change is the hallmark of a concave up graph.

85. What are the coordinates of the vertex of the quadratic $f(x) = (x - 4)^{2} - 9$?
    • $(-4, -9)$
    • $(4, 9)$
    • $(4, -9)$
    • $(-4, 9)$

    Correct answer: $(4, -9)$

    The vertex is $(4, -9)$. In vertex form $a(x - h)^{2} + k$, the vertex is the point $(h, k)$; here $h = 4$ and $k = -9$. The subtraction inside the squared term means the horizontal coordinate is positive 4, not -4, which the incorrect choices reverse.

86. The graph of $g(x) = \frac{1}{x - 3} + 2$ is the reciprocal function $f(x) = \frac{1}{x}$ transformed how?
    • Shifted left 3 and down 2, with asymptotes $x = -3$ and $y = -2$
    • Shifted right 3 and up 2, with asymptotes $x = 3$ and $y = 2$
    • Shifted right 2 and up 3, with asymptotes $x = 2$ and $y = 3$
    • Shifted left 3 and up 2, with asymptotes $x = -3$ and $y = 2$

    Correct answer: Shifted right 3 and up 2, with asymptotes $x = 3$ and $y = 2$

    Right 3 and up 2 is correct. Replacing x with $x - 3$ moves the graph right 3, sending the vertical asymptote to $x = 3$, and adding 2 outside moves it up 2, sending the horizontal asymptote to $y = 2$. The inside subtraction shifts the graph in the positive x-direction, so a leftward shift does not apply.

87. In the factored polynomial $f(x) = (x + 1)(x - 2)(x - 2)(x + 1)$, what is the multiplicity of the zero at $x = 2$ and how does the graph behave there?
    • Multiplicity 1; the graph crosses the x-axis
    • Multiplicity 4; the graph crosses the x-axis
    • Multiplicity 3; the graph flattens and crosses
    • Multiplicity 2; the graph touches the x-axis and turns around

    Correct answer: Multiplicity 2; the graph touches the x-axis and turns around

    Multiplicity 2 with a touch-and-turn is correct. The factor $(x - 2)$ appears twice, giving the zero at $x = 2$ a multiplicity of 2, and an even multiplicity causes the graph to touch the x-axis and bounce back without crossing. The factor $(x + 1)$ also appears twice but governs the separate zero at $x = -1$.

88. What is the value of $\log_{6} 36$?
    • 6
    • 2
    • 3
    • 12

    Correct answer: 2

    The value is 2. The expression asks for the power to which 6 must be raised to produce 36, and since $6^{2}$ equals 36, the answer is 2. The other choices do not satisfy 6 raised to that power equaling 36.

89. What is the value of $\log_{2}\frac{1}{8}$?
    • -3
    • 3
    • -8
    • $\frac{1}{3}$

    Correct answer: -3

    The value is -3. Since one-eighth equals $2^{-3}$, the logarithm returns that exponent, giving -3. A positive answer would correspond to a value greater than 1, not a fraction less than 1.

90. An exponential function passes through $(2, 18)$ and $(3, 54)$. What is its base?
    • 2
    • 6
    • 3
    • 9

    Correct answer: 3

    The base is 3. The base equals the ratio of consecutive outputs over a unit increase in input, so 54 divided by 18 gives 3. The base is not the difference of the outputs or either output value itself.

91. Solve for x: $5^{x} = 125$.
    • $x = 25$
    • $x = 5$
    • $x = 4$
    • $x = 3$

    Correct answer: $x = 3$

    $x = 3$ is correct. Rewriting 125 as $5^{3}$ makes the bases match, so the exponents are equal and x equals 3. Dividing or taking a root does not isolate the exponent here.

92. Solve for x: $3^{2x} = 81$.
    • $x = 2$
    • $x = 4$
    • $x = 8$
    • $x = 3$

    Correct answer: $x = 2$

    $x = 2$ is correct. Since 81 equals $3^{4}$, the exponents must be equal, so 2x equals 4 and x equals 2. Forgetting to divide by the coefficient of x would give the incorrect value 4.

93. What is the value of $\log_{9} 3$?
    • $\frac{1}{2}$
    • 3
    • 2
    • $\frac{1}{3}$

    Correct answer: $\frac{1}{2}$

    The value is one-half. Since 9 raised to the one-half power equals $\sqrt{9}$, which is 3, the logarithm equals one-half. A whole-number answer would require 3 to be a whole-number power of 9, which it is not.

94. Rewrite the exponential statement $4^{3} = 64$ in logarithmic form.
    • $\log_{3} 64 = 4$
    • $\log_{4} 64 = 3$
    • $\log_{64} 4 = 3$
    • $\log_{4} 3 = 64$

    Correct answer: $\log_{4} 64 = 3$

    $\log_{4} 64 = 3$ is correct. The base of the power becomes the base of the logarithm, the result becomes the argument, and the exponent becomes the logarithm's value. The other forms swap the roles of these parts incorrectly.

95. What is the value of $e^{\ln 5}$?
    • $\ln 5$
    • 1
    • 5
    • E

    Correct answer: 5

    The value is 5. Because the natural exponential and natural logarithm are inverse functions, raising e to the natural log of a number returns that number, so the result is 5. The two functions undo each other entirely.

96. What is the value of $\ln(e^{4})$?
    • 4
    • 1
    • E
    • $e^{4}$

    Correct answer: 4

    The value is 4. Since the natural logarithm is the inverse of the natural exponential, taking ln of e to a power returns that exponent, giving 4. The base and exponent are not multiplied together.

97. Which transformation does $g(x) = 3 \times 2^{x}$ apply to $f(x) = 2^{x}$?
    • A horizontal shift right by 3
    • A vertical shift up by 3
    • A reflection over the y-axis
    • A vertical stretch by a factor of 3

    Correct answer: A vertical stretch by a factor of 3

    A vertical stretch by a factor of 3 is correct. Multiplying the entire function by 3 scales every output by 3, stretching the graph vertically away from the x-axis. A shift would require adding a constant, not multiplying.

98. Which transformation does $g(x) = 2^{-x}$ apply to $f(x) = 2^{x}$?
    • A reflection over the x-axis
    • A vertical shift down
    • A reflection over the y-axis
    • A horizontal stretch

    Correct answer: A reflection over the y-axis

    A reflection over the y-axis is correct. Replacing x with negative x flips the graph horizontally across the y-axis, turning a growth curve into a decay curve. Negating the entire output, not the input, would instead reflect over the x-axis.

99. Expand $\log_{b}\left(\frac{x}{y}\right)^{3}$ using the properties of logarithms.
    • $3(\log_{b} x - \log_{b} y)$
    • $\log_{b} x - 3\log_{b} y$
    • $3\log_{b} x - \log_{b} y$
    • $3(\log_{b} x \times \log_{b} y)$

    Correct answer: $3(\log_{b} x - \log_{b} y)$

    $3(\log_{b} x - \log_{b} y)$ is correct. The outer exponent 3 moves out front by the power property, and the quotient inside becomes a difference of logarithms that the 3 distributes across. The coefficient must apply to both terms, not just one.

100. Condense into a single logarithm: $\log_{b} 8 + 2\log_{b} x$.
     • $\log_{b}(8 + x^{2})$
     • $\log_{b}(16x)$
     • $\log_{b}(8x^{2})$
     • $\log_{b}(8 \times 2x)$

     Correct answer: $\log_{b}(8x^{2})$

     $\log_{b}(8x^{2})$ is correct. The coefficient 2 becomes an exponent on x, and adding two logarithms combines them into the logarithm of a product, giving 8 times $x^{2}$. Adding the arguments instead of multiplying them is invalid.

101. What is the value of $\log_{8} 2$?
     • $\frac{1}{3}$
     • 3
     • 4
     • $\frac{1}{4}$

     Correct answer: $\frac{1}{3}$

     The value is one-third. Since 8 raised to the one-third power is $\sqrt[3]{8}$, which equals 2, the logarithm equals one-third. A whole-number answer would require 2 to be a whole-number power of 8.

102. A bacteria culture follows $N(t) = 400 \times 3^{t}$, where t is in hours. How many bacteria are present at $t = 2$ hours?
     • 2400
     • 1200
     • 3600
     • 1800

     Correct answer: 3600

     The value is 3600. Substituting $t = 2$ gives 400 times $3^{2}$, which is 400 times 9, equal to 3600. Multiplying the initial value by the base only once would incorrectly give the one-hour value.

103. Express the change of base for $\log_{5} 20$ using natural logarithms.
     • $\frac{\ln 20}{\ln 5}$
     • $\ln 20 \times \ln 5$
     • $\frac{\ln 5}{\ln 20}$
     • $\ln 20 - \ln 5$

     Correct answer: $\frac{\ln 20}{\ln 5}$

     $\frac{\ln 20}{\ln 5}$ is correct. The change of base formula places the logarithm of the argument over the logarithm of the original base, both in the new base, so it is ln 20 over ln 5. Reversing the ratio would compute a different logarithm.

104. What is the y-intercept of $f(x) = 7 \times \left(\frac{1}{2}\right)^{x}$?
     • $\frac{1}{2}$
     • 0
     • 1
     • 7

     Correct answer: 7

     The y-intercept is 7. Evaluating at $x = 0$ gives 7 times a base raised to the zero power, which is 7 times 1, so the graph crosses the y-axis at 7. The base does not affect the y-intercept because any base to the zero power is 1.

105. Solve for x: $2^{x + 1} = 32$.
     • $x = 4$
     • $x = 6$
     • $x = 5$
     • $x = 16$

     Correct answer: $x = 4$

     $x = 4$ is correct. Since 32 equals $2^{5}$, the exponents must match, so x plus 1 equals 5 and x equals 4. Forgetting to subtract 1 from 5 would give the incorrect value 5.

106. If a quantity is multiplied by 4 every time the input increases by 1, what is the base of the exponential model?
     • $\frac{1}{4}$
     • 0
     • 4
     • 1

     Correct answer: 4

     The base is 4. The base of an exponential function is the constant factor applied for each unit increase in the input, which is 4 here. A base less than 1 would indicate decay rather than this growth by a factor of 4.

107. Which value satisfies $\log_{x} 49 = 2$?
     • $x = 24.5$
     • $x = 49$
     • $x = 7$
     • $x = 14$

     Correct answer: $x = 7$

     $x = 7$ is correct. Rewriting in exponential form gives $x^{2}$ equals 49, so x equals 7 since the base must be positive. The base, not the argument, is what is being solved for here.

108. A loan balance is modeled by $B(t) = 10000 \times (1.05)^{t}$. What is the annual growth rate?
     • 5 percent
     • 1.05 percent
     • 50 percent
     • 105 percent

     Correct answer: 5 percent

     The annual growth rate is 5 percent. The base 1.05 equals 1 plus 0.05, so the rate of increase each year is 0.05, or 5 percent. The base itself is the growth factor, not the rate.

109. What is the value of $\log_{100} 10$?
     • 2
     • $\frac{1}{2}$
     • 10
     • $\frac{1}{10}$

     Correct answer: $\frac{1}{2}$

     The value is one-half. Since 100 raised to the one-half power is $\sqrt{100}$, which equals 10, the logarithm equals one-half. A value of 2 would instead describe log base 10 of 100.

110. An exponential decay function has the form $f(x) = a \cdot b^{x}$. Which describes the typical relationship between consecutive outputs?
     • Each output is a constant amount less than the previous
     • Each output equals the previous output
     • Each output is a fixed fraction of the previous output
     • Each output is $\sqrt{\text{the previous}}$

     Correct answer: Each output is a fixed fraction of the previous output

     A fixed fraction of the previous output is correct. With a base between 0 and 1, multiplying by that base each step makes each output a constant fraction of the one before it. A constant subtracted amount would describe linear decay instead.

111. Solve for x: $\log_{5}(2x) = 2$.
     • $x = 5$
     • $x = 12.5$
     • $x = 10$
     • $x = 25$

     Correct answer: $x = 12.5$

     $x = 12.5$ is correct. Converting to exponential form gives $5^{2}$ equals 2x, so 25 equals 2x and x equals 12.5. Forgetting to divide by the coefficient 2 would leave the incorrect value 25.

112. Which statement correctly compares the steepness of $f(x) = 2^{x}$ and $g(x) = 5^{x}$ for positive x?
     • Both rise at exactly the same rate
     • Neither rises for positive x
     • G rises more steeply than f because its base is larger
     • F rises more steeply than g because its base is smaller

     Correct answer: G rises more steeply than f because its base is larger

     g rises more steeply is correct. A larger base multiplies each output by a bigger factor per unit, so the function with base 5 climbs faster than the one with base 2 for positive inputs. A smaller base produces gentler growth, not steeper.

113. What is the value of $\log_{4}\frac{1}{16}$?
     • -2
     • -4
     • 2
     • $\frac{1}{2}$

     Correct answer: -2

     The value is -2. Since one-sixteenth equals $4^{-2}$, the logarithm returns that exponent, giving -2. Arguments less than 1 produce negative logarithms when the base is greater than 1.

114. A model gives the temperature of a cooling object as $T(t) = 20 + 60 \times (0.8)^{t}$. What value does the temperature approach as t grows large?
     • 0
     • 60
     • 80
     • 20

     Correct answer: 20

     The temperature approaches 20. As t grows, the term 60 times $0.8^{t}$ decays toward zero, leaving the constant 20, which is the horizontal asymptote of the model. The constant added outside the exponential sets this long-run value.

115. Which expression is equivalent to $\log_{b}(\sqrt{x})$?
     • $2\log_{b} x$
     • $\frac{\log_{b} x}{x}$
     • $(\log_{b} x)^{2}$
     • $\frac{1}{2}\log_{b} x$

     Correct answer: $\frac{1}{2}\log_{b} x$

     $\frac{1}{2}\log_{b} x$ is correct. A square root is the one-half power, and the power property moves that exponent out front as a coefficient of one-half. Treating the root as a multiplier of 2 would reverse the operation.

116. What is the value of $\log_{2} 1 + \log_{2} 16$?
     • 16
     • 5
     • 0
     • 4

     Correct answer: 4

     The value is 4. The logarithm of 1 is 0 in any base, and $\log_{2} 16$ is 4 since $2^{4}$ equals 16, so the sum is 0 plus 4, equal to 4. The first term contributes nothing because the log of 1 is always zero.

117. Solve for x: $e^{2x} = e^{6}$.
     • $x = 6$
     • $x = 12$
     • $x = 4$
     • $x = 3$

     Correct answer: $x = 3$

     $x = 3$ is correct. With equal bases of e, the exponents must be equal, so 2x equals 6 and x equals 3. Forgetting to divide the right side by 2 would give the incorrect value 6.

118. A savings account earns interest so that the balance triples every 4 years, modeled by $A = P \times 3^{t/4}$. What does the divisor 4 represent?
     • The initial deposit
     • The number of years for the balance to triple
     • The annual interest rate
     • The final balance

     Correct answer: The number of years for the balance to triple

     The tripling time is correct. The divisor 4 in the exponent means the base 3 is fully applied once every 4 years, so 4 is the time required for the balance to triple. The divisor sets the length of each tripling period.

119. Which is the correct exponential form of the statement $\ln(7) = k$?
     • $e^{7} = k$
     • $10^{k} = 7$
     • $k^{e} = 7$
     • $e^{k} = 7$

     Correct answer: $e^{k} = 7$

     $e^{k} = 7$ is correct. The natural logarithm uses base e, so $\ln(7) = k$ converts to e raised to the k power equaling 7. Using base 10 would describe the common logarithm, not the natural one.

120. What is the value of $\log_{5} 5^{3}$?
     • 125
     • 3
     • 15
     • 5

     Correct answer: 3

     The value is 3. Because the logarithm and exponential share the base 5 and are inverses, $\log_{5} 5^{3}$ returns the exponent 3. The base and exponent are not multiplied to give 15.

121. A drug's concentration in the blood follows $C(t) = 50 \times (0.5)^{t/6}$ milligrams. What is the half-life of the drug?
     • 50 hours
     • 6 hours
     • 0.5 hours
     • 12 hours

     Correct answer: 6 hours

     The half-life is 6 hours. With base one-half and exponent $t/6$, the concentration is multiplied by one-half once every 6 hours, so the half-life is 6 hours. The divisor in the exponent sets this halving period.

122. Which property allows $\ln(a \times b)$ to be rewritten?
     • $\ln a \times \ln b$
     • $\ln a + \ln b$
     • $\frac{\ln a}{\ln b}$
     • $\ln a - \ln b$

     Correct answer: $\ln a + \ln b$

     $\ln a + \ln b$ is correct. The product property of logarithms applies to natural logarithms as well, so the log of a product is the sum of the logs. Multiplying or dividing the logarithms is not a valid identity.

123. For the function $f(x) = a \cdot b^{x}$, what does the parameter $b$ control?
     • The y-intercept of the graph
     • The location of the vertical asymptote
     • The horizontal asymptote
     • Whether the function grows or decays and how quickly

     Correct answer: Whether the function grows or decays and how quickly

     Controlling growth or decay and its rate is correct. The base $b$ determines if outputs increase or decrease and the speed of that change, since it is the per-step multiplier. The initial value $a$ sets the y-intercept, a separate feature.

124. What is the value of $4^{\log_{4} 9}$?
     • 36
     • 4
     • 2
     • 9

     Correct answer: 9

     The value is 9. Raising a base to the logarithm of a number in that same base returns the number, since the two operations are inverses, so the result is 9. The base and argument are not multiplied together.

125. Solve for x: $\log_{6} x = 0$.
     • $x = 6$
     • $x = 1$
     • $x = 0$
     • $x = 6^{6}$

     Correct answer: $x = 1$

     $x = 1$ is correct. Converting to exponential form gives $6^{0} = x$, and any nonzero base to the zero power is 1, so $x = 1$. The logarithm of 1 is 0 in every base.

126. If $f(x) = b^{x}$ has a base $b$ between 0 and 1, what is the end behavior as x increases without bound?
     • The outputs increase without bound
     • The outputs approach 0
     • The outputs approach 1
     • The outputs become negative

     Correct answer: The outputs approach 0

     Approaching 0 is correct. A base between 0 and 1 means repeated multiplication by a fraction, so the positive outputs shrink toward the asymptote $y = 0$ as x grows. The outputs stay positive because a positive base to any power is positive.

127. An arithmetic sequence has first term 7 and common difference 4. What is the explicit formula for its nth term?
     • $a_n = 7 + 4(n - 1)$
     • $a_n = 7 \times 4^{n-1}$
     • $a_n = 4 + 7(n - 1)$
     • $a_n = 7 + 4n$

     Correct answer: $a_n = 7 + 4(n - 1)$

     The formula $a_n = 7 + 4(n - 1)$ is correct. An arithmetic sequence's nth term is the first term plus the common difference times one less than the term number, here $7 + 4(n - 1)$. Using a multiplicative factor would describe a geometric sequence instead.

128. A geometric sequence has first term 5 and common ratio 3. What is its explicit formula for the nth term?
     • $a_n = 5 + 3(n - 1)$
     • $a_n = 5 \times 3^{n-1}$
     • $a_n = 3 \times 5^{n-1}$
     • $a_n = 5 \times 3^{n}$

     Correct answer: $a_n = 5 \times 3^{n-1}$

     The formula $a_n = 5 \times 3^{n-1}$ is correct. A geometric sequence's nth term is the first term times the common ratio raised to one less than the term number. Adding the ratio instead would give an arithmetic sequence.

129. An arithmetic sequence corresponds to which type of function when the term number is the input?
     • An exponential function
     • A linear function
     • A quadratic function
     • A logarithmic function

     Correct answer: A linear function

     A linear function is correct. Because an arithmetic sequence adds a constant common difference each step, its terms change by a constant additive amount, matching the constant slope of a linear function. A constant multiplicative change would correspond to an exponential function.

130. The fourth term of a geometric sequence is 54 and the common ratio is 3. What is the first term?
     • 2
     • 6
     • 18
     • 3

     Correct answer: 2

     The first term is 2. The fourth term equals the first term times the ratio cubed, so 54 equals first term times 27, giving a first term of 2. Dividing 54 by 27 isolates the first term.

131. For the exponential function $f(x) = 7 \times \left(\frac{1}{4}\right)^{x}$, by what factor does the output change each time x increases by 1?
     • It is multiplied by $\frac{1}{4}$
     • It is multiplied by 7
     • It decreases by 4
     • It is multiplied by 4

     Correct answer: It is multiplied by $\frac{1}{4}$

     Multiplying by $\frac{1}{4}$ is correct. The base of an exponential function is the factor applied for each unit increase in the input, here $\frac{1}{4}$. The coefficient 7 is the initial value, not the per-step factor.

132. If an exponential function has a growth factor of 1.5 per unit, what is the equivalent percent rate of change per unit?
     • A 50 percent increase
     • A 150 percent increase
     • A 1.5 percent increase
     • A 15 percent increase

     Correct answer: A 50 percent increase

     A 50 percent increase is correct. A growth factor of 1.5 equals 1 plus 0.50, so the quantity grows by 50 percent each unit. The factor 1.5 itself is not the percent; the percent comes from the part beyond 1.

133. An exponential function f satisfies $f(0) = 6$ and has a constant ratio of 2 per unit increase in x. What is $f(3)$?
     • 48
     • 12
     • 36
     • 24

     Correct answer: 48

     The value is 48. Starting at 6 and doubling for each of three unit steps gives $6 \times 2^{3}$, which is $6 \times 8 = 48$. The ratio applies multiplicatively once per unit step.

134. Which describes the additive transformation property of an exponential function $f(x) = b^{x}$?
     • $f(x + k) = f(x) + f(k)$
     • $f(x + k) = f(x) \times b^{k}$
     • $f(x + k) = f(x) + k$
     • $f(x + k) = \frac{f(x)}{b^{k}}$

     Correct answer: $f(x + k) = f(x) \times b^{k}$

     $f(x + k) = f(x) \times b^{k}$ is correct. Adding k to the input multiplies the output by $b^{k}$, because exponents add when powers of the same base multiply. This multiplicative response to additive input is the defining feature of exponentials.

135. What is the value of $27^{2/3}$?
     • 9
     • 18
     • 6
     • 3

     Correct answer: 9

     The value is 9. A fractional exponent means take $\sqrt[3]{27}$, which is 3, then square it to get 9. The denominator gives the root and the numerator gives the power.

136. What is the value of $16^{-1/2}$?
     • $\frac{1}{4}$
     • 4
     • -4
     • $\frac{1}{16}$

     Correct answer: $\frac{1}{4}$

     The value is $\frac{1}{4}$. The exponent $-1/2$ means take $\sqrt{16}$, which is 4, then take the reciprocal because of the negative sign, giving $\frac{1}{4}$. The negative exponent inverts rather than negates.

137. Which expression is equivalent to the $\sqrt{b}$ expressed with an exponent?
     • $b^{1/2}$
     • $b^{2}$
     • $b^{-1/2}$
     • $2^{b}$

     Correct answer: $b^{1/2}$

     $b^{1/2}$ is correct. A square root is the same as raising to the one-half power, since squaring that result returns b. A negative exponent would give a reciprocal instead.

138. Which expression equals $(a \times b)^{n}$?
     • $a^{n} \times b^{n}$
     • $a^{n} + b^{n}$
     • $a \times b^{n}$
     • $(a + b)^{n}$

     Correct answer: $a^{n} \times b^{n}$

     $a^{n} \times b^{n}$ is correct. The power of a product distributes the exponent to each factor, so $(a \times b)^{n}$ equals $a^{n} \times b^{n}$. Adding the powers is not a valid rule.

139. What is the value of $\log_{2}\left(\frac{1}{8}\right)$?
     • -3
     • 3
     • $-\frac{1}{3}$
     • $\frac{1}{8}$

     Correct answer: -3

     The value is -3. Since $2^{-3} = \frac{1}{8}$, the logarithm is -3. A negative logarithm signals that the argument is a fraction between zero and one for a base greater than one.

140. Why is $\log_{5}(-25)$ undefined over the real numbers?
     • No real power of 5 produces a negative number
     • Because 25 is too large
     • Because the base must be negative
     • Because logarithms require even arguments

     Correct answer: No real power of 5 produces a negative number

     No real power of a positive base is negative, so the expression is undefined. A positive base raised to any real exponent yields a positive result, so it can never equal -25. This is why logarithm arguments must be positive.

141. Using the product property, $\log_{b} 6$ can be rewritten as which sum?
     • $\log_{b} 2 + \log_{b} 3$
     • $\log_{b} 2 \times \log_{b} 3$
     • $\log_{b} 4 + \log_{b} 2$
     • $\log_{b} 3 - \log_{b} 2$

     Correct answer: $\log_{b} 2 + \log_{b} 3$

     $\log_{b} 2 + \log_{b} 3$ is correct. Because 6 equals 2 times 3, the product property turns the logarithm of the product into the sum of the logarithms of the factors. The factors must multiply to 6, which 2 and 3 do.

142. Expand $\log_{b}(x^{2} \times y)$ using logarithm properties.
     • $2 \log_{b} x + \log_{b} y$
     • $2 \log_{b} x \times \log_{b} y$
     • $\log_{b} x + 2 \log_{b} y$
     • $2 (\log_{b} x + \log_{b} y)$

     Correct answer: $2 \log_{b} x + \log_{b} y$

     $2 \log_{b} x + \log_{b} y$ is correct. The product becomes a sum of logarithms, and the exponent 2 on x moves out front as a multiplier by the power property. The exponent applies only to x, not to y.

143. Expand $\log_{b}\left(\frac{x}{y^{3}}\right)$ using logarithm properties.
     • $\log_{b} x - 3 \log_{b} y$
     • $\log_{b} x + 3 \log_{b} y$
     • $3 \log_{b} x - \log_{b} y$
     • $\log_{b} x - \log_{b} 3y$

     Correct answer: $\log_{b} x - 3 \log_{b} y$

     $\log_{b} x - 3 \log_{b} y$ is correct. The quotient becomes a difference of logarithms, and the exponent 3 on y moves out as a multiplier on its logarithm. Subtraction reflects the division and the multiplier reflects the power.

144. Use the change of base formula to write $\log_{5} 20$ with natural logarithms.
     • $\frac{\ln(20)}{\ln(5)}$
     • $\frac{\ln(5)}{\ln(20)}$
     • $\ln(20) \times \ln(5)$
     • $\ln(20) - \ln(5)$

     Correct answer: $\frac{\ln(20)}{\ln(5)}$

     $\frac{\ln(20)}{\ln(5)}$ is correct. The change of base formula puts the argument's logarithm over the base's logarithm in any common base, so the argument 20 goes on top and the base 5 on the bottom. Reversing the ratio would compute the wrong value.

145. Solve for x: $3^{x} = 81$.
     • $x = 4$
     • $x = 3$
     • $x = 27$
     • $x = 9$

     Correct answer: $x = 4$

     $x = 4$ is correct. Since 81 equals $3^{4}$, the equation becomes $3^{x} = 3^{4}$, so the exponents match and $x = 4$. Rewriting both sides with the same base makes the exponents directly comparable.

146. Solve for x: $2^{x + 1} = 32$.
     • $x = 4$
     • $x = 5$
     • $x = 6$
     • $x = 16$

     Correct answer: $x = 4$

     $x = 4$ is correct. Writing 32 as $2^{5}$ gives $2^{x + 1} = 2^{5}$, so $x + 1 = 5$ and $x = 4$. The exponents must be equal once both sides share the same base.

147. Solve for x exactly: $e^{x} = 10$.
     • $x = \ln(10)$
     • $x = 10/e$
     • $x = \log_{10} e$
     • $x = e^{10}$

     Correct answer: $x = \ln(10)$

     $x = \ln(10)$ is correct. Taking the natural logarithm of both sides undoes the base e, leaving x equal to $\ln(10)$. The natural logarithm is the inverse of the natural exponential function.

148. Solve for x: $5^{x} = 17$, expressed with logarithms.
     • $x = \log_{5} 17$
     • $x = 17/5$
     • $x = \log_{17} 5$
     • $x = 5 \times \log 17$

     Correct answer: $x = \log_{5} 17$

     $x = \log_{5} 17$ is correct. Converting the exponential equation to logarithmic form, the exponent equals the logarithm of the result in the equation's base. The base 5 stays the base of the logarithm and 17 becomes the argument.

149. Solve for x: $\log_{4}(2x) = 2$.
     • $x = 8$
     • $x = 4$
     • $x = 16$
     • $x = 2$

     Correct answer: $x = 8$

     $x = 8$ is correct. Converting to exponential form gives $4^{2} = 2x$, so $2x = 16$ and $x = 8$. After rewriting in exponential form, solve the resulting linear equation.

150. Solve for x: $\ln(x) = 3$, expressed exactly.
     • $x = e^{3}$
     • $x = 3e$
     • $x = \ln(3)$
     • $x = 3/e$

     Correct answer: $x = e^{3}$

     $x = e^{3}$ is correct. Converting $\ln(x) = 3$ to exponential form raises the base e to the power 3, giving $x = e^{3}$. The natural logarithm and the base e exponential are inverse operations.

151. Solve for x: $\log_{6}(x) + \log_{6}(x - 5) = \log_{6} 24$.
     • $x = 8$
     • $x = 3$
     • $x = -3$
     • $x = 24$

     Correct answer: $x = 8$

     $x = 8$ is correct. Combining the left side with the product property gives $\log_{6}[x(x - 5)] = \log_{6} 24$, so $x(x - 5) = 24$; solving $x^{2} - 5x - 24 = 0$ gives $x = 8$, rejecting the negative since the logarithm needs positive arguments.

152. Why must solutions to a logarithmic equation be checked in the original equation?
     • Some algebraic solutions can make a logarithm's argument zero or negative
     • Logarithms always have two solutions
     • Checking changes the value of the solution
     • Logarithmic equations never have valid solutions

     Correct answer: Some algebraic solutions can make a logarithm's argument zero or negative

     Checking for invalid arguments is correct. Algebraic steps may introduce extraneous solutions that make a logarithm's argument zero or negative, which is outside the domain, so each candidate must be verified. Only solutions keeping all arguments positive are valid.

153. The function $g(x) = 3 \times 2^{x}$ is which transformation of $f(x) = 2^{x}$?
     • A vertical stretch by a factor of 3
     • A horizontal shift right by 3
     • A vertical shift up by 3
     • A reflection over the y-axis

     Correct answer: A vertical stretch by a factor of 3

     A vertical stretch by a factor of 3 is correct. Multiplying the exponential by 3 scales every output away from the x-axis by that factor, stretching the graph vertically. The horizontal asymptote at $y = 0$ stays in place.

154. The function $g(x) = 2^{-x}$ is which transformation of $f(x) = 2^{x}$?
     • A reflection over the y-axis
     • A reflection over the x-axis
     • A vertical shift down
     • A horizontal shift left

     Correct answer: A reflection over the y-axis

     A reflection over the y-axis is correct. Replacing x with -x flips the graph horizontally across the y-axis, converting growth into decay. The outputs stay positive, so it is not a reflection over the x-axis.

155. The graph of $g(x) = \log_{2}(x - 3)$ is the graph of $\log_{2} x$ shifted how?
     • Right by 3 units
     • Left by 3 units
     • Up by 3 units
     • Down by 3 units

     Correct answer: Right by 3 units

     Right by 3 units is correct. Subtracting 3 inside the logarithm shifts the graph to the right by 3, moving the vertical asymptote from $x = 0$ to $x = 3$. Inside changes move the graph horizontally, opposite to the sign.

156. After the transformation $g(x) = 2^{x} - 5$, where is the horizontal asymptote?
     • $y = -5$
     • $y = 0$
     • $y = 5$
     • $x = -5$

     Correct answer: $y = -5$

     $y = -5$ is correct. Subtracting 5 from the exponential output shifts the entire graph, including its horizontal asymptote, down by 5 units from $y = 0$ to $y = -5$. The asymptote moves with the vertical shift.

157. Find the equation of an exponential function passing through (0, 8) and (2, 2).
     • $f(x) = 8 \times \left(\frac{1}{2}\right)^{x}$
     • $f(x) = 8 \times 2^{x}$
     • $f(x) = 2 \times \left(\frac{1}{2}\right)^{x}$
     • $f(x) = 8 \times \left(\frac{1}{4}\right)^{x}$

     Correct answer: $f(x) = 8 \times \left(\frac{1}{2}\right)^{x}$

     $f(x) = 8 \times \left(\frac{1}{2}\right)^{x}$ is correct. The initial value is 8 from the first point, and the base squared must equal 2 divided by 8, which is $\frac{1}{4}$, so the base is $\frac{1}{2}$. The base is found from the ratio of outputs over the input change.

158. A bacteria culture triples every 4 hours and starts at 200 cells. Which model gives the count after t hours?
     • $P = 200 \times 3^{t/4}$
     • $P = 200 \times 3^{4t}$
     • $P = 200 \times 4^{t/3}$
     • $P = 200 + 3t$

     Correct answer: $P = 200 \times 3^{t/4}$

     $P = 200 \times 3^{t/4}$ is correct. Tripling every 4 hours means the base 3 is applied once each 4-hour period, so the exponent is t divided by 4. The divisor sets how often the growth factor is applied.

159. A sample of 60 mg of a substance has a half-life of 8 years. Which model gives the amount after t years?
     • $A = 60 \times \left(\frac{1}{2}\right)^{t/8}$
     • $A = 60 \times \left(\frac{1}{2}\right)^{8t}$
     • $A = 60 \times 2^{t/8}$
     • $A = 60 - \left(\frac{1}{8}\right)t$

     Correct answer: $A = 60 \times \left(\frac{1}{2}\right)^{t/8}$

     $A = 60 \times \left(\frac{1}{2}\right)^{t/8}$ is correct. A half-life of 8 years means the quantity is multiplied by one-half once every 8 years, so the exponent is t divided by 8 with base one-half. The divisor is the half-life.

160. An account earns 6 percent interest compounded annually on an initial 1000 dollars. Which model gives the balance after t years?
     • $A = 1000 \times (1.06)^{t}$
     • $A = 1000 \times (0.06)^{t}$
     • $A = 1000 \times (0.94)^{t}$
     • $A = 1000 + 60t$

     Correct answer: $A = 1000 \times (1.06)^{t}$

     $A = 1000 \times (1.06)^{t}$ is correct. Annual growth of 6 percent multiplies the balance by 1 plus 0.06 = 1.06 each year, applied to the initial 1000 dollars. A base of 0.06 or a linear form would not represent compound growth.

161. Approximately how many years does it take a quantity growing at a continuous rate of 7 percent to double, using the rule based on natural logarithms?
     • About 10 years, since $\frac{\ln(2)}{0.07}$ is roughly 9.9
     • About 2 years
     • About 50 years
     • About 14 years

     Correct answer: About 10 years, since $\frac{\ln(2)}{0.07}$ is roughly 9.9

     About 10 years is correct. Doubling time for continuous growth equals $\ln(2)$ divided by the rate, and $\ln(2)$ is about 0.69, so $\frac{0.69}{0.07}$ is roughly 9.9 years. The natural logarithm of 2 governs continuous doubling time.

162. Which best describes how a logarithmic function grows for very large inputs?
     • It keeps increasing but grows slower than any positive power of x
     • It approaches a horizontal asymptote
     • It eventually decreases
     • It grows faster than exponential functions

     Correct answer: It keeps increasing but grows slower than any positive power of x

     Slow unbounded growth is correct. A logarithmic function increases without bound but more slowly than any positive power of the input, so it never levels off yet is outpaced by polynomial and exponential growth. It has no horizontal asymptote.

163. The decibel level is a logarithmic measure of sound intensity. If intensity increases by a factor of 100, how does this relate to a logarithmic scale based on powers of 10?
     • It is an increase of 2 on the base-10 log scale
     • It is an increase of 100 on the log scale
     • It is an increase of 10 on the log scale
     • It does not change the log value

     Correct answer: It is an increase of 2 on the base-10 log scale

     An increase of 2 is correct. A factor of 100 equals $10^{2}$, so on a base-10 logarithmic scale the value rises by 2. Multiplying the underlying quantity by a power of ten adds that exponent on the log scale.

164. Data plotted on a graph with a logarithmic vertical axis falls along a straight line. What does this indicate about the underlying relationship?
     • The data follows an exponential model
     • The data follows a linear model
     • The data follows a logarithmic model
     • The data has no pattern

     Correct answer: The data follows an exponential model

     An exponential model is correct. When the logarithm of the outputs is linear in the inputs, the original relationship is exponential, which is exactly what a straight line on a semi-log plot reveals. This is the standard diagnostic for exponential data.

165. What is the value of the composition $\log_{3}(3^{7})$?
     • 7
     • 21
     • 3
     • 10

     Correct answer: 7

     The value is 7. The logarithm and the exponential of the same base are inverses, so composing them returns the original exponent, here 7. The base and exponent are not multiplied together.

166. Which expression gives the exact solution of the equation $5^{x} = 20$?
     • $x = \frac{\log 20}{\log 5}$
     • $x = \frac{\log 5}{\log 20}$
     • $x = \frac{20}{5}$
     • $x = \log 20 - \log 5$

     Correct answer: $x = \frac{\log 20}{\log 5}$

     $x = \frac{\log 20}{\log 5}$ is correct. Taking the logarithm of both sides gives $x \log 5 = \log 20$, and dividing both sides by $\log 5$ isolates x as the quotient $\frac{\log 20}{\log 5}$. Reversing the ratio or subtracting the logarithms does not solve for the exponent.

167. What is the value of $\tan\frac{\pi}{4}$?
     • $\frac{1}{\sqrt{3}}$
     • 1
     • 0
     • $\sqrt{3}$

     Correct answer: 1

     The value is 1. At $\frac{\pi}{4}$ ($45^{\circ}$) the sine and cosine are both $\frac{\sqrt{2}}{2}$, so their ratio, the tangent, equals 1. Equal sine and cosine always give a tangent of 1.

168. What is the value of $\cos\frac{\pi}{4}$?
     • $\frac{\sqrt{2}}{2}$
     • $\frac{\sqrt{3}}{2}$
     • 1
     • $\frac{1}{2}$

     Correct answer: $\frac{\sqrt{2}}{2}$

     The value is $\frac{\sqrt{2}}{2}$. At $\frac{\pi}{4}$ ($45^{\circ}$) the unit circle point is $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$, and the cosine is the x-coordinate. The sine there is the same value.

169. What is the value of $\cos\frac{\pi}{2}$?
     • -1
     • 1
     • 0
     • $\frac{\sqrt{2}}{2}$

     Correct answer: 0

     The value is 0. At $\frac{\pi}{2}$ ($90^{\circ}$) the unit circle point is $(0, 1)$, and the cosine equals the x-coordinate, which is 0. The sine at this angle is 1.

170. What is the value of $\sin\pi$?
     • $\frac{\sqrt{2}}{2}$
     • -1
     • 1
     • 0

     Correct answer: 0

     The value is 0. At an angle of $\pi$ ($180^{\circ}$) the unit circle point is $(-1, 0)$, and the sine equals the y-coordinate, which is 0. The cosine there is -1.

171. What is the value of $\sin\frac{3\pi}{2}$?
     • $\frac{\sqrt{3}}{2}$
     • 1
     • 0
     • -1

     Correct answer: -1

     The value is -1. At $\frac{3\pi}{2}$ ($270^{\circ}$) the unit circle point is $(0, -1)$, and the sine equals the y-coordinate, which is -1. The cosine there is 0.

172. What is the value of $\tan\frac{\pi}{3}$?
     • $\frac{1}{2}$
     • $\sqrt{3}$
     • 1
     • $\frac{\sqrt{3}}{3}$

     Correct answer: $\sqrt{3}$

     The value is $\sqrt{3}$. At $\frac{\pi}{3}$ the sine is $\frac{\sqrt{3}}{2}$ and the cosine is $\frac{1}{2}$, so the tangent, sine over cosine, equals $\frac{\sqrt{3}/2}{1/2} = \sqrt{3}$.

173. What is the value of $\arcsin 1$?
     • $\pi$
     • $\frac{\pi}{2}$
     • $\frac{\pi}{4}$
     • 0

     Correct answer: $\frac{\pi}{2}$

     The value is $\frac{\pi}{2}$. Arcsine returns the angle in the restricted range whose sine equals the input, and $\sin\frac{\pi}{2}$ is 1. This is the largest output arcsine can give.

174. What is the value of $\arccos 0$?
     • $\frac{\pi}{4}$
     • 0
     • $\pi$
     • $\frac{\pi}{2}$

     Correct answer: $\frac{\pi}{2}$

     The value is $\frac{\pi}{2}$. Arccosine returns the angle in its restricted range whose cosine equals the input, and $\cos\frac{\pi}{2}$ is 0. So $\arccos 0$ is $\frac{\pi}{2}$.

175. What is the value of $\arctan 1$?
     • $\frac{\pi}{4}$
     • 0
     • $\frac{\pi}{2}$
     • $\pi$

     Correct answer: $\frac{\pi}{4}$

     The value is $\frac{\pi}{4}$. Arctangent returns the angle in its restricted range whose tangent equals the input, and $\tan\frac{\pi}{4}$ is 1. So $\arctan 1$ is $\frac{\pi}{4}$.

176. What is the range of the cosine function?
     • From 0 to 1
     • From -2 to 2
     • From -1 to 1, inclusive
     • All real numbers

     Correct answer: From -1 to 1, inclusive

     From -1 to 1 is correct. Since the cosine equals the x-coordinate of a point on the unit circle, its values stay between -1 and 1. This matches the range of the sine function.

177. What is the domain of the tangent function?
     • Only positive numbers
     • All real numbers
     • From -1 to 1
     • All real numbers except where cosine is zero

     Correct answer: All real numbers except where cosine is zero

     All reals except where cosine is zero is correct. Tangent is sine divided by cosine, so it is undefined at every angle where the cosine equals zero, such as $\frac{\pi}{2}$ and its odd multiples. Everywhere else it is defined.

178. What is the range of the tangent function?
     • From $-\pi$ to $\pi$
     • From 0 to $\infty$
     • All real numbers
     • From -1 to 1

     Correct answer: All real numbers

     All real numbers is correct. As an angle approaches a vertical asymptote, the tangent grows without bound in both directions, so it attains every real value. Its range is therefore all real numbers, unlike sine and cosine.

179. Which polar equation produces a four-petaled rose?
     • $\theta = 4$
     • $r = \cos(2\theta)$
     • $r = 4\cos\theta$
     • $r = 4$

     Correct answer: $r = \cos(2\theta)$

     $r = \cos(2\theta)$ is correct. Rose curves have the form $r$ equals a cosine or sine of $(n\theta)$; when n is even the rose has 2n petals, so n = 2 yields four petals. A constant r gives a circle instead.

180. A polar equation of the form $r = a + a\cos\theta$ traces what named curve?
     • A circle centered at the origin
     • A spiral
     • A cardioid
     • A straight line

     Correct answer: A cardioid

     A cardioid is correct. When the constant and the coefficient of cosine are equal, the limacon becomes a heart-shaped cardioid that passes through the origin. Different ratios produce other limacon shapes.

181. The polar curve $r = 1 + 2\cos\theta$ is an example of which family of curves?
     • A parabola
     • A line
     • A limacon
     • A rose

     Correct answer: A limacon

     A limacon is correct. Equations of the form $r = a + b\cos\theta$ (or $\sin\theta$) are limacons; when b is larger than a, the curve forms an inner loop. Roses instead use a multiple of theta inside the trig function.

182. For the polar function $r = f(\theta)$, how is the rate of change of distance from the origin interpreted?
     • It is always zero
     • It tells how fast r changes as the angle $\theta$ changes
     • It equals the area enclosed by the curve
     • It gives the period of the curve

     Correct answer: It tells how fast r changes as the angle $\theta$ changes

     Rate of change of r with theta is correct. In a polar graph the rate of change of f describes how quickly the radial distance grows or shrinks as the angle increases. A positive rate means moving away from the origin; a negative rate means moving toward it.

183. On a polar graph, an interval where r is positive and decreasing means the curve is doing what?
     • Moving away from the origin
     • Staying at a fixed distance
     • Moving toward the origin while still on the same side as $\theta$
     • Crossing to the opposite side of the origin

     Correct answer: Moving toward the origin while still on the same side as $\theta$

     Moving toward the origin is correct. A positive r places the point in the direction of theta, and a decreasing r shrinks that distance, so the curve approaches the origin while rotating. The point stays on the theta side because r remains positive.

184. Which equation converts rectangular coordinates to the polar angle $\theta$?
     • $\theta = \sqrt{x^{2} + y^{2}}$
     • $\theta = \arctan\left(\frac{y}{x}\right)$
     • $\theta = x \times y$
     • $\theta = x + y$

     Correct answer: $\theta = \arctan\left(\frac{y}{x}\right)$

     Arctangent of y over x is correct. The angle is recovered from the ratio of the y- and x-coordinates using the inverse tangent, with quadrant adjustments as needed. The square root expression gives r, not theta.

185. A point has rectangular coordinates $(0, 3)$. What is a valid polar representation?
     • $\left(3, \frac{\pi}{2}\right)$
     • $(3, 0)$
     • $(0, 3)$
     • $(3, \pi)$

     Correct answer: $\left(3, \frac{\pi}{2}\right)$

     $\left(3, \frac{\pi}{2}\right)$ is correct. The point lies on the positive y-axis at distance 3 from the origin, so $r = 3$ and $\theta = \frac{\pi}{2}$. The radial distance is $\sqrt{0^{2} + 3^{2}} = 3$.

186. Which identity correctly expresses sine squared theta using cosine?
     • $\sin^{2}\theta = \cos\theta$
     • $\sin^{2}\theta = 1 - \cos^{2}\theta$
     • $\sin^{2}\theta = 1 + \cos^{2}\theta$
     • $\sin^{2}\theta = \cos^{2}\theta - 1$

     Correct answer: $\sin^{2}\theta = 1 - \cos^{2}\theta$

     $1 - \cos^{2}\theta$ is correct. Rearranging the Pythagorean identity $\sin^{2}\theta + \cos^{2}\theta = 1$ isolates sine squared as $1 - \cos^{2}\theta$. The other forms break the identity.

187. If $\cos\theta = -\frac{4}{5}$ and $\theta$ is in the second quadrant, what is $\sin\theta$?
     • $-\frac{3}{5}$
     • $\frac{3}{5}$
     • $\frac{4}{5}$
     • $-\frac{4}{5}$

     Correct answer: $\frac{3}{5}$

     $\sin\theta = \frac{3}{5}$ is correct. From the Pythagorean identity, $\sin^{2}\theta = 1 - \frac{16}{25} = \frac{9}{25}$, so $\sin\theta = \pm\frac{3}{5}$; in the second quadrant the sine is positive, giving $\frac{3}{5}$.

188. In which quadrant is the tangent function positive while the sine is negative?
     • Quadrant III
     • Quadrant I
     • Quadrant IV
     • Quadrant II

     Correct answer: Quadrant III

     Quadrant III is correct. There both sine and cosine are negative, so their ratio the tangent is positive, while the sine itself remains negative. This matches the requested combination of signs.

189. What is the reference angle for an angle of $\frac{5\pi}{6}$?
     • $\frac{\pi}{2}$
     • $\frac{\pi}{6}$
     • $\frac{5\pi}{6}$
     • $\frac{\pi}{3}$

     Correct answer: $\frac{\pi}{6}$

     The reference angle is $\frac{\pi}{6}$. The angle $\frac{5\pi}{6}$ lies in the second quadrant, and its reference angle is $\pi - \frac{5\pi}{6} = \frac{\pi}{6}$, the acute angle to the x-axis.

190. What is the reference angle for an angle of $\frac{4\pi}{3}$?
     • $\frac{\pi}{3}$
     • $\frac{\pi}{6}$
     • $\frac{4\pi}{3}$
     • $\frac{2\pi}{3}$

     Correct answer: $\frac{\pi}{3}$

     The reference angle is $\frac{\pi}{3}$. The angle $\frac{4\pi}{3}$ is in the third quadrant, and its reference angle is $\frac{4\pi}{3} - \pi = \frac{\pi}{3}$, the acute angle to the x-axis.

191. An angle of $-\frac{\pi}{4}$ is coterminal with which positive angle?
     • $\frac{7\pi}{4}$
     • $\frac{5\pi}{4}$
     • $\frac{3\pi}{4}$
     • $\frac{\pi}{4}$

     Correct answer: $\frac{7\pi}{4}$

     $\frac{7\pi}{4}$ is correct. Adding a full rotation of $2\pi$ to $-\frac{\pi}{4}$ gives $-\frac{\pi}{4} + 2\pi = \frac{7\pi}{4}$, which shares the same terminal side. Coterminal angles differ by whole rotations.

192. What is the length of an arc subtended by a central angle of $\frac{\pi}{3}$ in a circle of radius 6?
     • $3\pi$
     • $\frac{\pi}{2}$
     • $2\pi$
     • $6\pi$

     Correct answer: $2\pi$

     The arc length is $2\pi$. Using $s = r\theta$ with $r = 6$ and $\theta = \frac{\pi}{3}$ gives $s = 6 \times \frac{\pi}{3} = 2\pi$. The angle must be in radians for this formula.

193. What is $270^{\circ}$ expressed in radians?
     • $\frac{\pi}{2}$ radians
     • $\frac{3\pi}{2}$ radians
     • $3\pi$ radians
     • $\frac{2\pi}{3}$ radians

     Correct answer: $\frac{3\pi}{2}$ radians

     $\frac{3\pi}{2}$ radians is correct. Multiplying $270^{\circ}$ by $\frac{\pi}{180}$ gives $\frac{270\pi}{180} = \frac{3\pi}{2}$ radians. This is three-quarters of a full circle.

194. An angle of $\frac{2\pi}{3}$ radians equals how many degrees?
     • $60^{\circ}$
     • $150^{\circ}$
     • $120^{\circ}$
     • $90^{\circ}$

     Correct answer: $120^{\circ}$

     $120^{\circ}$ is correct. Multiplying $\frac{2\pi}{3}$ by $\frac{180}{\pi}$ gives $\frac{2 \times 180}{3} = \frac{360}{3} = 120^{\circ}$. The pi values cancel during conversion.

195. The graph of $f(x) = \sin\left(x - \frac{\pi}{3}\right)$ is the sine graph shifted how?
     • Down by $\frac{\pi}{3}$
     • Left by $\frac{\pi}{3}$
     • Right by $\frac{\pi}{3}$
     • Up by $\frac{\pi}{3}$

     Correct answer: Right by $\frac{\pi}{3}$

     Shifted right by $\frac{\pi}{3}$ is correct. Subtracting a positive constant inside the function argument translates the graph to the right by that amount. This is the phase shift of the sinusoid.

196. For $f(x) = 4\sin x - 2$, what is the maximum value?
     • 6
     • 4
     • 2
     • -6

     Correct answer: 2

     The maximum is 2. The amplitude is 4 and the midline is -2, so the peak is one amplitude above the midline at -2 plus 4 = 2. The minimum would be -2 minus 4 = -6.

197. For $f(x) = 4\sin x - 2$, what is the minimum value?
     • -6
     • 2
     • -4
     • -2

     Correct answer: -6

     The minimum is -6. With amplitude 4 and midline -2, the lowest point sits one amplitude below the midline at -2 minus 4 = -6. The maximum is -2 plus 4 = 2.

198. A sinusoid has a maximum of 12 and a minimum of -4. What is its amplitude?
     • 8
     • 4
     • 16
     • 12

     Correct answer: 8

     The amplitude is 8. Amplitude equals half the difference of the maximum and minimum, so (12 minus (-4)) divided by 2 = 16 divided by 2 = 8. The midline would be the average, which is 4.

199. A sinusoid has a maximum of 12 and a minimum of -4. What is its midline?
     • 0
     • 16
     • 8
     • 4

     Correct answer: 4

     The midline is 4. The midline is the average of the maximum and minimum, so (12 plus (-4)) divided by 2 = 8 divided by 2 = 4. The amplitude, the half-difference, is 8.

200. What is the period of $f(x) = \sin(3x)$?
     • $6\pi$
     • $\frac{\pi}{3}$
     • $3\pi$
     • $\frac{2\pi}{3}$

     Correct answer: $\frac{2\pi}{3}$

     The period is $\frac{2\pi}{3}$. The period equals $2\pi$ divided by the input coefficient, so with a coefficient of 3 the period is $\frac{2\pi}{3}$. A larger coefficient shortens the period.

201. Which describes the relationship between the secant graph and the cosine graph?
     • Secant has vertical asymptotes where cosine equals zero and shares cosine's sign
     • Secant has the same range as cosine
     • The secant graph crosses zero wherever cosine does
     • Secant is the cosine graph shifted down

     Correct answer: Secant has vertical asymptotes where cosine equals zero and shares cosine's sign

     Asymptotes where cosine is zero is correct. Since secant is the reciprocal of cosine, it blows up to $\infty$ exactly where cosine equals zero and takes the same sign as cosine elsewhere. Its range excludes the interval between $-1$ and $1$.

202. What is the value of $\sec 0$?
     • Undefined
     • -1
     • 0
     • 1

     Correct answer: 1

     The value is 1. Secant is $1$ divided by cosine, and $\cos 0$ is 1, so $\sec 0$ is $1$ divided by $1 = 1$. Secant is undefined only where the cosine is zero.

203. What is the value of $\csc\frac{\pi}{2}$?
     • 0
     • -1
     • Undefined
     • 1

     Correct answer: 1

     The value is 1. Cosecant is $1$ divided by sine, and $\sin\frac{\pi}{2}$ is 1, so the cosecant is $1$ divided by $1 = 1$. Cosecant is undefined where the sine is zero.

204. Which sinusoidal function would best model a Ferris wheel where riders board at the lowest point and the height starts at its minimum?
     • A plain sine function with no shifts
     • A negative cosine function shifted up to the midline
     • A tangent function
     • A positive cosine function with a vertical shift

     Correct answer: A negative cosine function shifted up to the midline

     A negative cosine shifted up is correct. A negative cosine starts at its minimum at the input zero, matching boarding at the lowest point, and the vertical shift raises the midline to the wheel's center height. A plain sine starts at the midline instead.

205. What is the value of $\sin\left(-\frac{\pi}{2}\right)$?
     • 0
     • 1
     • Undefined
     • -1

     Correct answer: -1

     The value is -1. Because sine is an odd function, $\sin\left(-\frac{\pi}{2}\right)$ equals the negative of $\sin\frac{\pi}{2}$, which is $-1$. The terminal point is $(0, -1)$ on the unit circle.

206. What is the value of $\cos\left(-\frac{\pi}{3}\right)$?
     • $-\frac{1}{2}$
     • $\frac{1}{2}$
     • $-\frac{\sqrt{3}}{2}$
     • $\frac{\sqrt{3}}{2}$

     Correct answer: $\frac{1}{2}$

     The value is $\frac{1}{2}$. Because cosine is an even function, $\cos\left(-\frac{\pi}{3}\right)$ equals $\cos\frac{\pi}{3}$, which is $\frac{1}{2}$. The negative sign in the angle does not change an even function's output.

207. What is the value of $\cos\frac{3\pi}{2}$?
     • 0
     • -1
     • $\frac{\sqrt{2}}{2}$
     • 1

     Correct answer: 0

     The value is 0. At an angle of $\frac{3\pi}{2}$ (270 degrees) the unit circle point is $(0, -1)$, and the cosine equals the x-coordinate, which is 0. The sine there is $-1$.

208. An angle of 45 degrees is equivalent to how many radians?
     • $\frac{\pi}{6}$
     • $\frac{\pi}{3}$
     • $\frac{\pi}{4}$
     • $\frac{\pi}{2}$

     Correct answer: $\frac{\pi}{4}$

     The value is $\frac{\pi}{4}$. Multiplying 45 degrees by $\frac{\pi}{180}$ gives 45 pi over 180, which simplifies to $\frac{\pi}{4}$. This is the radian benchmark for a 45-degree angle.

209. An angle of $\frac{5\pi}{4}$ radians lies in which quadrant?
     • Quadrant III
     • Quadrant II
     • Quadrant I
     • Quadrant IV

     Correct answer: Quadrant III

     Quadrant III is correct. The angle $\frac{5\pi}{4}$ is between $\pi$ and $\frac{3\pi}{2}$, which spans the third quadrant. Both the sine and cosine are therefore negative at this angle.

210. What is the reference angle for an angle of $\frac{2\pi}{3}$?
     • $\frac{2\pi}{3}$
     • $\frac{\pi}{4}$
     • $\frac{\pi}{3}$
     • $\frac{\pi}{6}$

     Correct answer: $\frac{\pi}{3}$

     The reference angle is $\frac{\pi}{3}$. Since $\frac{2\pi}{3}$ lies in the second quadrant, its reference angle is $\pi$ minus $\frac{2\pi}{3}$, which equals $\frac{\pi}{3}$. The reference angle is the acute angle to the x-axis.

211. What is the value of $\cos\frac{2\pi}{3}$?
     • $\frac{1}{2}$
     • $-\frac{1}{2}$
     • $-\frac{\sqrt{3}}{2}$
     • $\frac{\sqrt{3}}{2}$

     Correct answer: $-\frac{1}{2}$

     The value is $-\frac{1}{2}$. The angle $\frac{2\pi}{3}$ is in the second quadrant with reference angle $\frac{\pi}{3}$, where cosine is $\frac{1}{2}$; cosine is negative in the second quadrant, so the value is $-\frac{1}{2}$.

212. What is the value of $\sin\frac{7\pi}{6}$?
     • $-\frac{1}{2}$
     • $-\frac{\sqrt{3}}{2}$
     • $\frac{1}{2}$
     • $\frac{\sqrt{3}}{2}$

     Correct answer: $-\frac{1}{2}$

     The value is $-\frac{1}{2}$. The angle $\frac{7\pi}{6}$ lies in the third quadrant with reference angle $\frac{\pi}{6}$, where sine is $\frac{1}{2}$; sine is negative in the third quadrant, so the value is $-\frac{1}{2}$.

213. The graph of the cosine function attains its maximum value of 1 at which input?
     • $x = \frac{\pi}{2}$
     • $x = \frac{3\pi}{2}$
     • $x = \pi$
     • $x = 0$

     Correct answer: $x = 0$

     The value $x = 0$ is correct. The cosine starts at its maximum of 1 when the input is 0, corresponding to the unit circle point $(1, 0)$. It returns to 1 again at every multiple of $2\pi$.

214. Over the interval from 0 to $\frac{\pi}{2}$, the sine function is doing what?
     • Constant
     • Oscillating between extremes
     • Increasing
     • Decreasing

     Correct answer: Increasing

     Increasing is correct. On the interval from 0 to $\frac{\pi}{2}$, the sine rises from 0 up to its maximum of 1. After $\frac{\pi}{2}$ it begins to decrease.

215. A sinusoidal graph is concave down near its maximum because the outputs there are doing what?
     • Decreasing at a decreasing rate toward the minimum
     • Increasing at an increasing rate
     • Changing from increasing to decreasing as the curve bends downward
     • Staying perfectly constant

     Correct answer: Changing from increasing to decreasing as the curve bends downward

     Bending downward at the peak is correct. Near a maximum a sinusoid is concave down, where the rate of change shifts from positive to negative and the curve turns over. Near a minimum, by contrast, the graph is concave up.

216. Where is the rate of change of the sine function greatest in magnitude?
     • Where it crosses the midline
     • At its maximum and minimum points
     • Where the amplitude is largest
     • Where the period ends

     Correct answer: Where it crosses the midline

     At the midline crossings is correct. A sinusoid changes fastest where it passes through the midline, since the graph is steepest there. At the maximum and minimum the rate of change is momentarily zero.

217. The cosecant function has vertical asymptotes wherever which function equals zero?
     • Tangent
     • Sine
     • Secant
     • Cosine

     Correct answer: Sine

     Where sine is zero is correct. Since cosecant is $1$ divided by sine, it is undefined and has vertical asymptotes exactly where the sine equals zero, such as at 0, $\pi$, and $2\pi$. These are different from the asymptotes of secant.

218. What is the range of the secant function?
     • All real numbers
     • All values less than or equal to $-1$ or greater than or equal to $1$
     • Only positive numbers
     • From $-1$ to $1$

     Correct answer: All values less than or equal to $-1$ or greater than or equal to $1$

     Values at least 1 in magnitude is correct. Because secant is the reciprocal of cosine and cosine ranges from $-1$ to $1$, the secant never lies strictly between $-1$ and $1$; its outputs are at least 1 in absolute value.

219. What is the range of the arcsine function (its output angles)?
     • All real numbers
     • From $-1$ to $1$
     • From $-\frac{\pi}{2}$ to $\frac{\pi}{2}$
     • From 0 to $\pi$

     Correct answer: From $-\frac{\pi}{2}$ to $\frac{\pi}{2}$

     From negative pi over 2 to pi over 2 is correct. Arcsine is defined to return angles only in the interval from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$, the restricted domain that makes sine one-to-one. This guarantees a single output angle.

220. What is the range of the arccosine function (its output angles)?
     • From 0 to $2\pi$
     • From $-\frac{\pi}{2}$ to $\frac{\pi}{2}$
     • From $-1$ to $1$
     • From 0 to $\pi$

     Correct answer: From 0 to $\pi$

     From 0 to pi is correct. Arccosine returns angles only in the interval from 0 to $\pi$, the restricted domain on which cosine is one-to-one. This ensures each input ratio maps to a single output angle.

221. The tangent function has vertical asymptotes that repeat with what spacing along the x-axis?
     • Every $2\pi$ units
     • They do not repeat
     • Every $\pi$ units
     • Every $\frac{\pi}{2}$ units

     Correct answer: Every $\pi$ units

     Every pi units is correct. The tangent's asymptotes occur where cosine is zero, at $\frac{\pi}{2}$ plus integer multiples of $\pi$, so consecutive asymptotes are $\pi$ apart. This matches the tangent's period of $\pi$.

222. Which polar equation produces a cardioid?
     • $r = 3$
     • $r = 2 + 2\cos\theta$
     • $r = 5\cos\theta$
     • $\theta = \frac{\pi}{4}$

     Correct answer: $r = 2 + 2\cos\theta$

     The equation $r = 2 + 2\cos\theta$ is correct. A cardioid is a limacon of the form $r = a + a\cos\theta$ where the constant and the coefficient are equal, producing a heart shape. A constant r gives a circle instead.

223. A polar equation of the form $r = a\sin(n\theta)$ with n a positive integer typically graphs as what?
     • A rose curve with petals
     • A parabola
     • A single circle centered at the origin
     • A straight line

     Correct answer: A rose curve with petals

     A rose curve is correct. Polar equations of the form $r = a\sin(n\theta)$ trace rose curves made of symmetric petals. The number of petals depends on whether n is odd or even.

224. For the rose curve $r = a\cos(n\theta)$, how many petals appear when n is an odd integer?
     • N petals
     • Always 4 petals
     • N divided by 2 petals
     • 2n petals

     Correct answer: N petals

     n petals is correct. When n is odd, the rose curve $r = a\cos(n\theta)$ has exactly n petals because the petals retrace over the same paths. When n is even, the curve instead has 2n petals.

225. How far from the origin is the point with polar coordinates $\left(6, \frac{5\pi}{4}\right)$?
     • 6 units
     • 12 units
     • 3 units
     • $\frac{5\pi}{4}$ units

     Correct answer: 6 units

     6 units is correct. The first coordinate in a polar pair is the radial distance r, so the point lies 6 units from the origin regardless of the angle. The angle $\frac{5\pi}{4}$ only sets the direction.

226. A point has rectangular coordinates $(0, -3)$. What is a polar representation?
     • $\left(3, \frac{\pi}{2}\right)$
     • $\left(3, \frac{3\pi}{2}\right)$
     • $(3, 0)$
     • $(3, \pi)$

     Correct answer: $\left(3, \frac{3\pi}{2}\right)$

     $\left(3, \frac{3\pi}{2}\right)$ is correct. The point lies on the negative y-axis at distance 3 from the origin, so r is 3 and the angle pointing straight down is $\frac{3\pi}{2}$. The angle $\frac{\pi}{2}$ would point straight up instead.

227. On the graph of a polar function $r = f(\theta)$, the function value r reaches a relative maximum where the curve does what?
     • Reaches its farthest distance from the origin along that direction
     • Has zero radius
     • Crosses the origin
     • Becomes a straight line

     Correct answer: Reaches its farthest distance from the origin along that direction

     Farthest from the origin is correct. A relative maximum of r corresponds to a point where the curve is locally farthest from the pole. Reading polar graphs means relating the value of r to distance from the origin.

228. For a polar function $r = f(\theta)$, where r is positive and decreasing as $\theta$ increases, the curve is doing what?
     • Staying at a constant distance
     • Moving away from the origin
     • Stopping completely
     • Moving toward the origin while rotating

     Correct answer: Moving toward the origin while rotating

     Moving toward the origin is correct. When r is positive but decreasing as $\theta$ grows, the points get closer to the pole while the angle continues to increase, so the curve spirals inward. Increasing r would move it outward.

229. Two angles are supplementary if their measures add to what?
     • $\frac{\pi}{4}$ radians
     • $\pi$ radians
     • $\frac{\pi}{2}$ radians
     • $2\pi$ radians

     Correct answer: $\pi$ radians

     pi radians is correct. Supplementary angles sum to a straight angle, which is $\pi$ radians or 180 degrees. Angles summing to $\frac{\pi}{2}$ would instead be complementary.

230. Which identity correctly relates tangent and secant?
     • $1 - \tan^{2}\theta = \sec^{2}\theta$
     • $1 + \tan^{2}\theta = \sec^{2}\theta$
     • $\tan\theta = \sec\theta$
     • $\tan^{2}\theta = \sec^{2}\theta$

     Correct answer: $1 + \tan^{2}\theta = \sec^{2}\theta$

     1 plus tangent squared equals secant squared is correct. Dividing the Pythagorean identity by cosine squared yields $1 + \tan^{2}\theta = \sec^{2}\theta$. This is one of the standard Pythagorean identities.

231. If $\tan\theta = \frac{3}{4}$ and $\theta$ is in the first quadrant, what is $\sin\theta$?
     • $\frac{5}{3}$
     • $\frac{3}{4}$
     • $\frac{4}{5}$
     • $\frac{3}{5}$

     Correct answer: $\frac{3}{5}$

     $\sin\theta = \frac{3}{5}$ is correct. With tangent equal to $\frac{3}{4}$ in the first quadrant, picture a right triangle with opposite 3 and adjacent 4, so the hypotenuse is 5 and sine equals opposite over hypotenuse, $\frac{3}{5}$. The cosine would be $\frac{4}{5}$.

232. What is the value of $\sin\left(-\frac{\pi}{6}\right)$?
     • $-\frac{1}{2}$
     • $\frac{\sqrt{3}}{2}$
     • $-\frac{\sqrt{3}}{2}$
     • $\frac{1}{2}$

     Correct answer: $-\frac{1}{2}$

     The value is $-\frac{1}{2}$. Because sine is an odd function, $\sin\left(-\frac{\pi}{6}\right)$ equals the negative of $\sin\frac{\pi}{6}$, and $\sin\frac{\pi}{6}$ is $\frac{1}{2}$, giving $-\frac{1}{2}$.

233. What is the value of $\cos\left(-\frac{\pi}{3}\right)$?
     • $-\frac{\sqrt{3}}{2}$
     • $\frac{1}{2}$
     • $-\frac{1}{2}$
     • $\frac{\sqrt{3}}{2}$

     Correct answer: $\frac{1}{2}$

     The value is $\frac{1}{2}$. Because cosine is an even function, $\cos\left(-\frac{\pi}{3}\right)$ equals $\cos\frac{\pi}{3}$, which is $\frac{1}{2}$. The even symmetry means the sign of the input does not change the output.

234. When fitting a sinusoidal regression model to periodic data, the technology produces which form?
     • An exponential equation
     • A quadratic equation
     • A model of the form $y = a\sin(bx + c) + d$
     • A linear equation

     Correct answer: A model of the form $y = a\sin(bx + c) + d$

     The sinusoidal form is correct. Sinusoidal regression estimates the parameters a, b, c, and d of a sine model to best fit periodic data. The result captures amplitude, period, phase shift, and midline of the trend.

235. In a sinusoidal model of hours of daylight over a year, which feature represents the difference between the longest and shortest days?
     • The midline
     • Twice the amplitude
     • The period
     • The phase shift

     Correct answer: Twice the amplitude

     Twice the amplitude is correct. The full swing from the maximum daylight to the minimum daylight equals the distance from peak to trough, which is two amplitudes. A single amplitude measures only from the midline to a peak.

236. The graph of $f(x) = \tan x$ passes through the origin and is increasing. Between two consecutive asymptotes, the tangent function does what?
     • Stays constant
     • Reaches a maximum then a minimum
     • Decreases throughout
     • Increases throughout from $-\infty$ to $+\infty$

     Correct answer: Increases throughout from $-\infty$ to $+\infty$

     Increasing across each branch is correct. On each interval between consecutive vertical asymptotes, the tangent rises continuously from $-\infty$ to $+\infty$. It has no maximum or minimum values.

237. A sinusoidal function crosses its midline at $x = 1$ going upward and next crosses the midline going upward at $x = 7$. What is its period?
     • 7
     • 12
     • 3
     • 6

     Correct answer: 6

     The period is 6. One full period is the distance between two successive midline crossings in the same direction, here from $x = 1$ to $x = 7$, a span of 6. A half-period would separate upward and downward crossings.

238. Which describes the graph of a polar equation $r = 1 + 3\cos\theta$?
     • A limacon with an inner loop
     • A circle of radius 1
     • A straight line
     • A rose with three petals

     Correct answer: A limacon with an inner loop

     A limacon with an inner loop is correct. For $r = a + b\cos\theta$, when the coefficient b is larger than the constant a, the curve is a limacon that forms an inner loop. When they are equal it becomes a cardioid instead.

239. What is the exact value of $\cos\frac{5\pi}{6}$?
     • $-\frac{1}{2}$
     • $-\frac{\sqrt{3}}{2}$
     • $\frac{1}{2}$
     • $\frac{\sqrt{3}}{2}$

     Correct answer: $-\frac{\sqrt{3}}{2}$

     The value is $-\frac{\sqrt{3}}{2}$. The angle $\frac{5\pi}{6}$ is in the second quadrant with reference angle $\frac{\pi}{6}$, where cosine is $\frac{\sqrt{3}}{2}$; cosine is negative in the second quadrant, giving the negative value.

240. When converting the rectangular point $(1, 1)$ to polar coordinates, which equation is used to find the angle $\theta$?
     • $\theta = \tan\frac{y}{x}$
     • $\theta = \arctan\frac{y}{x}$ (arctangent)
     • $\theta = \frac{x}{y}$
     • $\theta = \sqrt{x^{2} + y^{2}}$

     Correct answer: $\theta = \arctan\frac{y}{x}$ (arctangent)

     The arctangent of $\frac{y}{x}$ is correct. Because $\tan\theta$ equals $\frac{y}{x}$, the angle is recovered by applying the inverse tangent to that ratio, with the quadrant of the point checked to place $\theta$ correctly. The $\sqrt{x^{2} + y^{2}}$ gives the radial distance r, not the angle.